N A S A C O N T R A C T O R

R E P O R T

COCVI

N A S A C R - 2 3 7 9

SENSITIVITY ANALYSISOF TORSIONAL VIBRATIONCHARACTERISTICS OFHELICOPTER ROTOR BLADESPart I - Structural Dynamics Analysis

by Theodore Bratanow and Akin Ecer

Prepared by

UNIVERSITY OF WISCONSIN-MILWAUKEE

Milwaukee, Wis.

for Langley Research Center

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • MARCH 1974

For sale by the National Technical Information Service, Springfield, Virginia 22151

1. Report No.NASA CR-2379

SENSITIVITY ANALYSIS OF TORSTICS OF HELICOPTER ROTOR BL/DYNAMICS ANALYSIS

7. Author(s)

2. Government Accession No.

3IONAL VlBRATK

\DES - PART I.)N CHARACTERIS-

STRUCTURAL

THEODORE BRATANOW AND AKIN ECER

9. Performing Organization Name and Address

UNIVERSITY OF WISCONSIN - MILWAUKEEMILWAUKEE, WISCONSIN

12. Sponsoring Agency Name and Address

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONWASHINGTON, D.C. 20546

15. Supplementary Notes

TOPICAL REPORT

16. Abstract

3. Recipient's Catalog No.

5. Report Date

March 1971*

6. Performing Organization Code

8. Performing Organization Report No.

10. Work Unit No.

11. Contract or Grant No.

NGR 50-007-00113. Type of Report and Period Covered

CONTRACTOR REPORT

14. Spons"ing Agency Code

A THEORETICAL INVESTIGATION OF STRUCTURAL VIBRATION CHARACTERISTICS OF ROTORBLADES WAS CARRIED OUT. COUPLED EQUATIONS OF MOTION FOR FLAPWISE BENDING ANDTORSION WERE FORMULATED FOR ROTOR BLADES WITH NONCOLLINEAR ELASTIC AND MASS AXES.THE FINITE ELEMENT METHOD WAS APPLIED FOR A DETAILED REPRESENTATION OF BLADESTRUCTURAL PROPERTIES. COUPLED STRUCTURAL MASS AND STIFFNESS COEFFICIENTS WEREEVALUATED. THE RANGE OF VALIDITY OF A SET OF COUPLED EQUATIONS OF MOTION LINEARIZED

WITH RESPECT TO ECCENTRICITY BETWEEN ELASTIC AND MASS AXES WAS INVESTIGATED. THESENSITIVITY OF BLADE VIBRATION CHARACTERISTICS TO TORSION WERE EVALUATED BY VARYINGBLADE GEOMETRIC PROPERTIES, BOUNDARY CONDITIONS, AND ECCENTRICITIES BETWEEN MASSAND ELASTIC AXES.

• Jtif.it.-sj-* •;=?« : •i/iiC.'-iffr-.'rk-^.-ri? --uj.- ':17. Key Words (Suggested by Author(s))

ROTOR BLADE DYNAMICSFINITE ELEMENT METHODCOUPLED RESPONSETORSIONAL COUPLING

19. Security Ctassif. (of this report)

UNCLASSIFIED

18. Distribution Statement

UNCLASSIFIED - UNLIMITED

Cat.20. Security Classif. (of this page)

UNCLASSIFIED

21. No. of Pages

46

3222. Price

$3.00

CONTENTS

Page

SUMMARY 1

INTRODUCTION 1

SYMBOLS 2

THE ANALYSIS OF THE STRUCTURAL VIBRATION CHARACTERISTICSOF ROTOR BLADES 5

Method of Solution 5

Variation of Coupled Vibration Characteristics withEccentricities between Elastic and Mass Axes 5

Stability of the Set of Linearized Coupled Equationsof Motion of Rotor Blades 6

Coupled Vibration Analysis of a Sample Rotor Blade 9

Stability of the Houbolt and Brooks equationsfor the sample blade 9

Effect of blade support conditions 10

Effect of eccentricity between mass andelastic centers 10

Effect of rotational speed 10

Effect of blade structural properties oncoupled torsional behavior 10

Concluding remarks . 11

APPENDIX A - DERIVATION OF THE EQUATIONS OF MOTION OF AROTATING BLADE IN FLAPWISE BENDING AND TORSION 13

APPENDIX B - FINITE ELEMENT FORMULATION OF THE LINEARIZEDEQUATIONS OF HOUBOLT AND BROOKS 17

APPENDIX C - FINITE ELEMENT FORMULATION OF MASS ANDSTIFFNESS MATRICES FOR THE COUPLED EQUATIONS OF MOTION 20

Formulation of Mass and Stiffness Matricesfor Torsional Vibrations 20

iii

Formulation of the Mass and Stiffness Matricesfor Bending Vibrations 21

Derivation of the Coupled Mass Term (mx 6) 23

2 'Derivation of the Coupled Stiffness Term (fi mrx ) 23D

Determination of the Coupled Mass andStiffness Matrices 23

REFERENCES 26

TABLES 27

FIGURES 32

IV

TABLES

Table Page

1 CH-34 Blade Physical Properties. 27

2 Uncoupled Frequencies of CH-34 Blade in Bending andTorsion. 28

3 The Frequencies Corresponding to the Coupled Modes,Consisting Mainly of the First Mode in Bending andthe First Mode in Torsion. 28

4a Generalized Structural Mass (a.. = p. M p.)>ij *-i — *-gXQ = 0.0254 m. for Uncoupled Modes of CH-34 Blade. 29

4b Generalized Structural Stiffness CY-• = P_- JiR-)>

xQ = 0.0254 m. for Uncoupled Modes of CH-34 Blade. • 30

5 Uncoupled Generalized Structural Mass (a.)>

Critical Damping (B-)» an<3 Stiffness (Y-)

of CH-34 Blade. 31

ILLUSTRATIONS

Figure Page

1 Variation of Coupled Frequencies of Rotating Blades. 322 Variation of Coupled Frequencies of Rotating Blades. 323 Uncoupled Natural Vibration Modes of CH-34 Blade. 334 Variation of the Two Lowest Frequencies Corresponding

to a Mainly Bending Mode with Increasing Eccentricity. 345 Variation of the Lowest Frequency Corresponding to a

Mainly Torsional Mode with Increasing Eccentricity. 346 Variation of Lowest Bending Frequencies for a Hinged

Blade CCH-34) with Blade Rotational Velocity,(x =0.0 and 0.083). 35

7 Variation of Lowest Bending Frequencies for a CantileverBlade (CH-34) with Blade Rotational Velocity. 35

8 Variation of Lowest Bending Frequencies for a Cantileverand Hinged Blade (CH-34) with Blade Rotational Velocity. 36

9 Variation of the Torsional Rotations at the Blade Tipwith Eccentricity. 36

10 First Uncoupled Bending Mode for CH-34 and Fixed EndCH-34. 37

11 Torsional Component of First Coupled Mode for CH-34and Fixed End CH-34. 37

12 Interaction Between Mainly Torsional Modes and HigherBending Modes. 38

13 Variation of First Coupled Modes with 30% Increase inWeight Distribution (W), (CH-34). 39

14 Variation of Second Coupled Mode with 30% Increase inTorsional Inertia (CH-34). 40

15 Variation of Torsional Frequencies with 30% Increasein Torsional Inertia (CH-34). 40

16 Variation of First Coupled Mode with 30% Increase inTorsional Rigidity (CH-34). 41

17 Variation of Torsional Frequencies with 30% Increasein Torsional Rigidity (CH-34). 41

18 Nodal Forces and Displacements for a Blade Element inBending and Torsion. 42

VI

SENSITIVITY ANALYSIS OF TORSIONAL1 VIBRATIONCHARACTERISTICS OF HELICOPTER ROTOR BLADES

PART I - STRUCTURAL DYNAMICS ANALYSIS

By Theodore Bratanow* and Akin Ecer**University of Wisconsin-Milwaukee

SUMMARY

A theoretical investigation of structural vibration characteristics ofrotor blades was carried out. Coupled equations of motion for flapwise bend-ing and torsion were formulated for rotor blades with non-collinear elasticand mass axes. The finite element method was applied for a detailed repre-sentation of blade structural properties. Coupled structural mass and stiff-ness coefficients were evaluated. The range of validity of a set of coupledequations of motion linearized with respect to eccentricity between elasticand mass axes was. investigated. The sensitivity of blade vibration charac-teristics to torsion were evaluated by varying blade geometric properties,boundary conditions, and eccentricities between mass and elastic axes.

INTRODUCTION

A complete analysis of the dynamic response of helicopter rotor bladesunder actual flight conditions is difficult because of the three-dimensionalnature of both blade deformations and unsteady flow around the blades. High-er operational forward speed and maneuverability requirements for modern hel-icopters have further increased the complexity of the problem. As indicatedby theoretical and experimental studies, at higher tip-speed ratios theblade dynamic response characteristics become increasingly sensitive to tor-sional oscillation.

The objective of the conducted investigation was to evaluate the effectof torsional oscillations on blade dynamic response characteristics and toanalyze the sensitivity of torsional oscillations to blade aerodynamic,structural, and geometric properties. In the first part of the analysis,the coupled equations of torsional and bending motion of a rotating bladewith non-collinear elastic and mass axes were developed in finite elementform considering only structural parameters. The results from this formu-lation were compared with results from a system of differential equationsof motion derived by Houbolt and Brooks for the same problem [1]. The im-portance of the second order terms in eccentricity between mass and elasticaxes was established. The obtained overall results were also compared withresults obtained by Isakson and Eisley [2].

Professor, ** Assistant Professor, Department of Mechanics, College ofEngineering and Applied Science.

SYMBOLS

a. a scalari

V V Cij-

D., E., H. generalized coupled mass and stiffness termsJ J J (see Eq. (A13))

b. a scalar

e. a scalari2

El bending stiffness of a blade, N-m

f. a scalari

h vertical displacement of a blade at the elasticcenter, m.

h_ column vector representing bending displacements

hr vertical displacement of a blade at the center ofmass, m.

h_. eigenvector of Eq. (B3)

h. eigenfunction corresponding to the bending displacementcomponent of p_. , m.

2IQ torsional inertia of the rotor blade, N-sec.w

_!_ identity matrix

2JG torsional stiffness of a blade, N-m

j( total stiffness matrix

K. total stiffness matrix of an element—i

K stiffness matrix in bending-o

](_ coupled stiffness matrix

JC stiffness matrix in torsion

JC, stiffness matrix in torsion for a blade elementi

£ length of blade element, m.

2 2m mass of the blade per unit length, N-sec. /m

M total mass matrix

M. total mass matrix of an element—i

M.. mass matrix in bending

M_ coupled mass matrix

M_ mass matrix in torsion

M mass matrix in torsion for a blade elementi

P a scalar (see Eq. (A21))

P_ column vector defining nodal forces in bending

q a scalar (see Eq. (A21))

R rotor radius, m.

2S non-dimensional eccentricity, S = xfi ni/Ip.

S.. non-dimensional eccentricity (see Eq. (A21))

S? non-dimensional eccentricity (see Eq. (A21))

S, . generalized coupled bending damping in torsion,N-sec./rad

t time variable

T centrifugal force along the longitudinal axis of arotating blade, N

U total energy for coupled bending and torsion, N-m

UD total energy in bending, N-mD

UT total energy in torsion, N-m

x distance on the longitudinal axis of cantilever bladefrom the fixed end, m.

xfl eccentricity between the elastic and mass center, m.

Y a scalar

6 column vector representing bending and torsional~~ displacements of a blade

£ non-dimensionalized abscissa of a point on a blade element

6 pitch change due to blade torsion, rad

0_ column vector representing torsional rotations

0. torsional rotation of a blade at the i mode, rad

X square of the ratio of lowest torsional frequency tolowest bending frequency

n ratio of torsional rotations to bending displacements inthe vibration mode corresponding to the lowest frequency,rad/m.

£_ column vector representing bending displacements

a) coupled frequency of the blade, rad/sec.

ft angular velocity of a rotating blade, rad/sec.

w natural frequency of a blade in bending, rad/sec.B

to lowest coupled frequency corresponding to a predominantlyi bending mode, rad/sec.

a) natural frequency of a blade in torsion, rad/sec.i

oj lowest coupled frequency corresponding to a predominantlyi torsional mode, rad/sec.

Superscript

t transpose

THE ANALYSIS OF STRUCTURAL VIBRATION CHARACTERISTICSOF ROTOR BLADES

Method of Solution

The energy method was applied in formulating the coupled equations ofmotion for helicopter rotor blades in flapwise bending and torsion. Theformulation is similar to the one used by Garland [3] for analyzing non-rotating beams. The details of the mathematical analysis are given in Ap-pendix A.

The coupled vibration characteristics of a rotating blade can be anal-yzed by first calculating the generalized coupled mass, damping, and stiff-ness coefficients in Eq. (A13) and then solving Eqs. (All) and (A12). Acoupled set of equations of motion, (Bl) , was formulated as shown in Appen- .dix B. A discretized set of mass and stiffness matrices was developed infinite element form. The derivation of each of the matrices in the set ofmatrix differential equations is given in Appendix C.

A numerical method was applied for the solution in Eqs. (All) andCA12). The following characteristics were established:

• coupled structural vibration characteristics of rotor bladesin bending and torsion

• stability characteristics of the equations of Houbolt and Brooks.

Variation of Coupled Vibration Characteristics withEccentricity between Elastic and Mass Axes

The variation of blade frequencies with increasing eccentricity betweenelastic and mass axes can be analysed from Eq. (A21). This analysis is de-scribed in Appendix A. The variation of the frequencies of the sample bladefor a two-degree-of-freedom system is illustrated in figure 1. As can beseen, when the non-dimensionalized eccentricity parameter S increases, thelower of the two frequencies corresponding to a predominantly bending modedecreases slightly, while the torsional frequency increases.

A further consideration of figure 1 indicates the main features of thecoupled structural behavior of a rotating blade, which can be summarized asfollows:

a) for values of the eccentricity xfl equal to zero (S = 0) un-

coupled bending and torsional frequencies correspond to q = 1and q = X, respectively. As the coupling increases, the fre-quency corresponding to a predominantly bending mode decreases,while the torsional frequency increases. As can be seen fromthe left portion of the curve, the bending frequencies arebound between

2 2 24 > <•> > «B

(1)

b) the left portion of the curve can be approximated by astraight line as follows:

q ^ 1 - yS

The ratio of torsional rotations to bending displacements can be calculatedfrom Eqs. (A18) and (A19) as

2n =

Observing also that

2 2co - toBl m_

aiel | w? u>2 + S(ft2-u2) T61

2 2)T » u)1

(4)

and

the ratio of coupling between torsional and bending modes is

mo,

(5)

(6)

Considering the left portion of the curve in figure 1, one can deduce fromEq. (6) that the coupling effect increases linearly with increasing eccen-tricity.

Stability of the Set of Linearized CoupledEquations of Motion of Rotor Blades

A set of equations of motion describing the vibrational behavior of arotor blade has been developed by Houbolt and Brooks [1] in the followingform:

( it \ ii / i \ i / 2 '\ ' /" " \Elh - Th -[mxfl x.9 + m h + x.9 = 0

/ \ / \ e ) \ e /

JG6 1 ' + mxn2xQh' + £22T 6 + 16 + mx.h = 0O O O O

(7)

(8)

Eqs. (7) and (8) have been used by several investigators for calculatingdynamic response of rotor blades [2,4]. These equations represent a linear-ized form of Eqs. (All) and (A12) with respect to x • only the first order

terms of x being considered. The second order terms E. and H. correspond9 3 Jto the change in moment of inertia of the blade and change in the positionof the axial force due to x , respectively.

In this portion of the work the effect of linearization of Eqs. (7)and (8) was investigated in two steps:

a) stability of the system of linearized Eqs. (7) and (8)

b) comparative evaluation of approximations in Eqs. (All) and (A12).

The sensitivity of frequencies to eccentricity between elastic and mass axeswas calculated from Eqs. (7) and (8). Neglecting the higher order terms ofx in Eqs. (A18) and (A19), these equations can be written in matrix form as

follows:

v 1

Iex _£.

6 m

1

J

al6l

b1£1

= 0 (9)

The dynamic system represented by Eq. (9) can be defined in terms of massand stiffness matrices as

[K. - u M] 6^ = [0] (10)

The stability of the system represented by Eq. (10) can be determinedby ascertaining whether the mass and stiffness matrices are positive defin-ite. For Eq. (1) one can show that

2xQm

xn0:e

<• ,

m< i

2 2T DTl Bl

det [M] > 0 (H)

det [K] > 0 (12)

Since - > 1, U3)n

the rotor blade vibrations become unstable in the respective first torsion-al and bending modes, if

x m/- > 1. (14)0

The eigenvalues of Eq. (10) can be calculated from the following equation:

det T M-1K - u>2 l]= i [(.X-q) (1-q) - S (q-p)2 1 = 0 (15)[_ — — — j J.-&2 L ^ J

The non-dimensionalized eccentricity between mass and elastic axesrelated to the non-dimensionalized coupled frequency is shown in figure 2.Several conclusions about the coupled behavior can be derived from thecurve shown in figure 2. From a comparison of Eq. (A.21) and Eq. (15) itfollows that

S - (1-qKX-q) and s _ (X-q) (1-q)S - ^ S -l - (p-q) (p-1) 2 - (p-q) (p-q)

From Eq. (16) one can derive the relation

C17)

Thus, one can conclude that the solution of Eq. (15) is satisfactory for theleft portion of the curve in figure 2. However, there is an important dif-ference for the right portion of the curve, since S /S- is not close tounity.

The following conclusions can be deduced from the above results:

a) the values of the lower coupled frequencies obtained fromthe Houbolt and Brooks equations would be satisfactory

b) Eqs. (7) and (8) become unstable when there is a smallincrease in eccentricity between the mass and elastic axes.

Such results are significant for the numerical integration of the equa-tions of motion. Instability of these equations leads to large torsionaldisplacements and thus to the conclusion that the actual system is unstable.

Coupled. Vibration Analysis of a Sample Rotor Blade

A sample rotor blade was chosen to illustrate the coupled vibrationbehavior. The finite element method, described in Appendices B and C, wasapplied for the numerical analysis. The structural properties of the sam-ple blade are listed in table 1. The uncoupled frequencies shown in table 2,were calculated using the numerical procedure summarized in Appendices B andC. The mode shapes for uncoupled vibrations are illustrated in figure 3.The coupled torsional behavior of the sample blade was then analyzed.

Stability of the Houbolt and Brooks equations for the sample blade -The analysis described in Appendix B was applied for determining the stabil-ity of the Houbolt and Brooks equations; checking where the torsional modesbecome unstable. Considering only two-degrees-of-freedom for the sampleblade, one can evaluate the critical eccentricity from

2 Al°lX = -—• , X = 0.089 (18)

For a four-degree-of-freedom system (two in bending and two in torsion), aquadratic expression was derived from the determinant of Eq. (Bll) for de-termining the critical eccentricity. The quadratic equation is

A1A2D1D2 - X6 [ VC22D1 + C?2D2> + VC11D2 + C21D1} ]

r r - r rLir22 L2ri2

From Eq. (19) the value of the critical eccentricity is obtained as

XQ = 0.057 (20)wThe variation of coupled frequencies with increasing eccentricity was

computed from the procedure described in Appendices B and C. The obtainednumerical results for a two-degree-of-freedom system are shown in figures 4and 5 and listed in table 3. The variation of frequency with increasing ec-centricity agrees with the results predicted by Eqs. (16) and (17). Al-though the bending frequencies are quite accurate, the torsional frequenciescan show considerable deviation. The instability of the torsional mode, asindicated in figure 5, agrees with the results from Eq. (18). For a multi-degree-of-freedom system the critical eccentricity that causes instabilitywas found to be in the range of

0.050 < XQ < 0.061 (21)w

These results agree with the results obtained from the two-degree-of-freedomsystem in Eq. (19). Thus, for the sample blade the equations of Houbolt andBrooks may become unstable for moderate eccentricities. One has to be care-ful when using such equations as a numerical integration basis for the de-termination of the response of rotor blades. On the other hand, if desired,the second order terms in x can be included quite conveniently in the anal-

9ysis by using the finite element formulation.

Effect of blade support conditions - The sample blade was analyzedwith an assumed hinged support, which allows flapwise rotations. The cal-culated frequencies for the lowest modes and for various blade rotationalspeeds are presented in figure 6. The sample blade was also analyzed fora fixed-end boundary condition, in flapwise bending. The vibration fre-quencies for this case are presented in figure 7 and compared with the fre-quencies of a hinged blade as shown in figure 8.

The coupled mode shapes of both hinged and fixed blades are shown infigures 10 and 11. As the end-fixity of the blade against bending rotationis increased, the bending stiffness increases; thus, the frequencies becomecorrespondingly higher. The torsional component of the lowest coupled modeis also increased.

Effect of eccentricity between mass and elastic centers - As shown infigure 7, the.lowest coupled frequency is affected by increasing the eccen-tricity between mass and elastic axes for the fixed blade; however, thisfrequency is less affected for the hinged blade. Frequencies correspondingto bending and torsional modes are compared for different eccentricities asshown in figure 12. Such results agree with the previously described re-sults on the variation of frequencies with increasing eccentricity. Thetorsional component of the lowest coupled mode is plotted for different ec-centricities in figure 12. The results also show the linear relationshipbetween eccentricity and the degree of coupling observed in Eq. (6).

Effect of rotational speed - The variation of frequency with increas-ing blade rotational speed for different boundary conditions and varying ec-centricities is shown in figures 6, 7, 8, and 12. As can be seen from thesefigures, the square of the coupled frequency term is linearly proportionalto the square of the rotational velocity.

The torsional rotation at the blade tip is plotted in figure 9 forvarious eccentricities at different rotational speeds. From the plottedcurves one can observe that the torsional coupled motion increases linearlywith eccentricity. For constant eccentricities this motion is also propor-tional to the square of the blade rotational speed.

Effect of blade structural properties on coupled torsional behavior -It was shown that an approximate relationship, Eq. (20), existed for thecoupled motion of rotor blades in bending and torsion. The validity of thisrelationship was examined by varying each of the parameters in the equation.The sensitivity of the torsional coupling parameter n with respect to eachof the parameters was also checked for a multi-degree-of-freedom system.

10

When the distribution of the blade weight was uniformly increased by30%, it was found that the bending and torsional frequencies decreased; thefirst bending frequency by 1.5% and the first torsional frequency by 2.3%.In accordance with Eq. (A20), the increase of the m/I ratio resulted in an

oincreased coupling of the response. As shown in figure 13 the torsionaldisplacements increased by 30%. It appears then that the torsional couplingis quite sensitive to weight changes.

Increasing the torsional inertia by 30% resulted in no change in thelowest bending frequency; however, the torsional frequency decreased asshown in figure 15. As shown in figure 14, there was also an increase incoupled torsional rotation in the second mode; however, there was no effecton the first mode. Such results can also be obtained from Eq. (A20). Theincreased torsional inertia and the resulting decrease in torsional frequen-cy has a balancing effect; producing no increase in coupling. The effect ofdecreasing torsional frequency is significant for the modes above the firstbending mode and causes a net increase as shown in figure 14.

The effect of an increase in the torsional rigidity on the frequenciesand the mode shapes is presented in figures 16 and 17. Increasing the tor-sional rigidity by 30% led to a 25% reduction of the maximum torsional ro-tation in the first coupled mode. The lower coupled frequency was not af-fected, although the predominantly torsional frequencies increased. Theplotted results correspond to Eq. (A20); showing an increase in torsionalfrequency and a decrease in the torsional component of the lowest coupledmode.

Increasing the bending inertia by 30% resulted in no significantchange in the bending and torsional frequencies or in the respective modeshapes. The results confirm again the validity of Eq. (A20), since couplingis not affected by a variation of the bending inertia.

Concluding Remarks

The presented analysis extends the application of the equations ofmotion of Houbolt and Brooks for consideration of larger eccentricities be-tween rotor blade elastic and mass axes. The presented results indicate thesignificance of individual structural parameters in determining the sen-sitivity of coupled bending and torsional vibration characteristics.

The results obtained for the sample blade show that for a rigorousanalysis of vibration response characteristics, one should consider that:

the blade boundary conditions are important when determiningthe coupling ratio of torsional to bending displacements

. the degree of coupling is linearly proportional to the squareof the eccentricity between the elastic and mass axes

11

a more accurate representation of mass and stiffness propertiesis required for the coupled analysis of bending and torsion.

When a detailed representation of the significant parameters is available,the blade sensitivity analysis can be used as a means of improving bladevibration and stability characteristics.

12

APPENDIX A

DERIVATION OF THE EQUATIONS OF MOTION OF A ROTATINGBLADE IN FLAPWISE BENDING AND TORSION

The total energy of the blade, when coupling is not considered, con-sists of

fR1 I " 2 9 9 '9UD = i (Elh - u£ mh + Th ) dx (Al)D / I D

J 0

for bending motion and

fRUT = \ (Jce'2 - co2 iQe2 + fi2 iee

2) dx (A2)Jo

for torsional motion.

For the coupled motion of the blade the displacements at the center ofmass of the blade can be defined as

hr = h f xfi (A3)0 D

whereby Eq. (A3) is valid only for small values of 6.

The potential energy for the coupled motion can be expressed as

" 9 9 9 ' 9 ' 9 9 9 9 2U = 4 (Elh - co mlv + Thr + JG8 Z - u 1.6 + n I0O dx (A4)1 ' u u y D

Substituting Eqs. (Al), (A2), and (A3) in Eq. (A4), one can obtain:

-Rf f

2 2-» i2 f 2 ^ T 0

2 2 02 2 2 . .(COD - co ) mh + (co™ - co ) I.Q - co m6 x. - co m6xQhD I D D O

+ TxV2 + 2Tx.e'h' dx + U_ + UT (A5)O D D i

For the bending and torsional displacements one can assume the fol-lowing functions:

13

h = a.h.i i (A6)

6 = b.j(A7)

where the displacement functions h. and 0. represent eigenvectors of the

uncoupled Eqs. (Al) and (.A2). One can also write

Elh.i"7 2 2

T + Th.B i i'2

dx (A8)

and apply the principle of minimum energy to obtain

3a.i= 0

For torsional motion one can similarly write

8U,= 0

(A9)

(AID)

Substituting Eqs. (A6) and (A7) in Eq. (A5), and considering the re-sults of Eqs. (A9) and (A10), according to the principle of minimum energy

Ai - X6bj . . .. = 0 (All)

2 2b. (uu - u) ) D. - x.a.j T. J 6 i

where

Ej H. = 0 (A12)

A. = I mh. dx,i i '

T0.h. dx,J J

14 -s

mh.e. dx, D. = I.62 dx,i j 3 '

R

E. = I me2 dx, H. = ~ I T9.'2 dx (A13)2

It is now possible to calculate the generalized coefficients (A. , B. .,C.., D., E., H.) and to investigate the effects of the coupling. 1 -*

An approximate investigation can be carried out when one considers thefirst bending mode and the first torsional mode of a rotating blade; i.e.,for i = 1 and j = 1, Eqs. (All) and (A12) become

a^ - a,2) AI - xebl :..]•'11 *" *"<1 1 I " (.AJ-TV

D. - x a Q\ - u Cn - to x"E. + ftx^ = 0 (A15)1 0 1 1 1 o l 0 1

The lowest uncoupled modes in bending and torsion, respectively, of arotating blade of uniform cross section can be approximated as

(A16)

For a rotor blade of uniform cross section and uniform eccentricityalong the blade, Eqs. (A14) and (A15) become

(ui2 - io2) ma.e. + (ft2 - to2) mxj^f. = 0 (A18)D I 1 1 o i l

(ft2 - o>2) mxaa,e, + < (oo2 - a)2) T -H mx2 (ft2 - a)2) > b.f, = 0 (A19)

o i l j l ^ D D 1 1

From Eqs. (A18) and (A19) we can now write

15

2x m(A20)

JT2

\ BlThe expression (A20) can be represented as

S = (q-1) (q-A)bl Cp-q) (P-D

16

APPENDIX B

FINITE ELEMENT FORMULATION OF THE LINEARIZED EQUATIONS OFHOUBOLT AND BROOKS

A general solution to the equations of Houbolt and Brooks can be ob-tained by formulating the equations in matrix form and then applying thefinite element theory. These equations can be written in general form asfollows:

< - vf - 0 (Bl)

The formulation of the matrices in Eq. (Bl) is given in Appendix C.energy of the blade can be expressed in matrix form as follows:

Total

u = - a) + x.eVh +o -- -L — -- -L —

1* 2

_B— H— —v,—

(B2)

For the displacement vectors h and Q_ one can assume expressions of thefollowing form:

- = [0] h = a.h.— 1—1

9 = b.6.- J-J

(B3)

(B4)

Eq. (B2) can then be written as

U = - 1 a)2 a2 A. + 2x a .b .C. . + b2 [ 1 1 6 1 3 i j

h K T h . -•• 2x Q a .b .B. .i-i-T-i e i i

2D.l3 J J

CB5)

where

B.. = e . K _ h . ,ij -j-C-i'

C. . = 9.M,,h. ,ij -j-C-i'

CB6)

CB7)"

17

An application of the principle of minimum potential energy on Eq. (B5) to-gether with Eqs. (B3) and (B4) leads now to the following system of equa-tions:

[••,2

V. 3

2

2- U)

_

aiAi + X0bj

b . D . + xea.

B.. - u C.. = 0

B. . - to C. • = 0ij 13

(B8)

(B9)

Eqs. (B8) and (B9) can be written in matrix form as follows:

2 2> -a) )

2 2(to -oj )A

B2 2

al

a2

•

am

bl

b2

•

•bn

0

0

•

•

0

0

•

(BIO)

The terms o>R , (»)„ , A., B. ., G. ., and D. in Eq. (BIO) can be evaluated fori j 1 1J XJ J

an uncoupled system of equations. Eq. (BIO) can then be solved to obtaincoupled frequencies.

The stability of Eqs. (B8) and (B9) can be tested by determiningwhether or not the mass matrix is positive definite. In matrix notationthis can be shown as

18

xecn Xeci2

Xec2i

det = 0 (BH)

xecn Xec2i

X8C12

19

APPENDIX C

FINITE ELEMENT FORMULATION OF MASS AND STIFFNESS MATRICESFOR THE COUPLED EQUATIONS OF MOTION

The finite element method was used for a discretized representation ofthe mass and stiffness properties of the rotor blade. The coupled equationsof motion (Eqs. (7) and (8)) were written as a system of matrix differentialequations. To analyze the torsional vibrations, a constant strain elementwas used which requires a linear variation of displacement over the element,while for bending vibrations a quadratic displacement function was employed.

Formulation of Mass and Stiffness Matrices for Torsional Vibrations

A consistent mass matrix was used for the discretized representation ofthe inertia term in the equation of motion (Al). The derivation of the massmatrix for torsional vibrations will now be summarized. For the blade ele-ment in figure 18, with nodal torsional rotations 6 and 6^, the rotationat any point can be expressed as

e = [1-5 5] (Cl)

The torsional acceleration at any point can then be written as

dt'

d2e,

dt

The equivalent inertia forces at the nodes of a blade finite element canbe defined as a vector T using the principle of virtual work as follows:

CC2)

T = 1-5

5

[1-5 5] (C3)

or

20

T =2

1

'dVdt' (C4)

dt

The elemental torsional mass matrix can be written as

(C5)

The elemental stiffness matrix can be analyzed in two parts. The firstpart is the elastic stiffness matrix, while the second part is the matrixrepresenting the effect of the centrifugal forces along the blade axis. Fora linear variation of torsional displacements over a finite element, usingan approach similar to the derivation of the mass matrix, the effect of thecentrifugal forces can be represented by a geometric stiffness matrix. Thetorsional stiffness nratrix for a rotating blade element can then be writtenas

1 -1

-1 1

I- £ 2 1

1 2(C6)

Formulation of the Mass and Stiffness Matrices for Bending Vibrations

After defining the nodal force vector P_ and the nodal displacement vec-tor p_, as shown in figure 18, the vertical displacement, h, at any pointcan be expressed in terms of the nodal displacements as follows:

h = (3C2 -

(C7)

A consistent mass matrix can also be derived for the bending vibrations ofthe finite element using the approach applied for torsional vibrations.Neglecting the effect of the shear deformations, the mass matrix in bendingcan be written as

21

m £420

156

22£

54

-m

4£

134

-3£2

Symmetric

156

-22£ 44'

(C8)

In the case of a rotating blade the bending stiffness can also be analyzedin two parts as elastic stiffness and geometric stiffness due to the axialforces in the blade. The representation of the elastic stiffness of a bladeelement is well known [5], The effect of rotation of the blade can be eval-uated by making use of a geometric stiffness matrix; which has already beenused for solving stability problems [5].

From Eq. (Al), the following relationship can be written for the finiteelement to represent the virtual work by axial forces:

h' Th' dx (C9)

Substituting Eq. (C7) into Eq. (C9), the bending stiffness of the blade ele-ment due to axial forces can be calculated. The total stiffness matrix inbending can then be written as

6EI

£2

6EI

4EI£

Symmetric

2EI£

-6EI 4EI

+ T£

65

£

-65

£10

2

-£10

-£2

30

* (•*> r. j. •Y?r TT— Symmetric

6_5

-£ 2£To is"

(CIO)

The displacement functions were defined fora blade element in bending,Eq. (C7), and for torsion, Eq, (Cl). Using the principle of virtual work,the coupling terms in Eqs. (7) and (8) can be derived for the defined dis-placement functions.

22

Derivation of the Coupled Mass Term (mx 6)

The virtual work due to the coupled mass term can be expressed as

P'P- h m x,6 dx (Cll)

..Substituting Eqs. (C2) and (C7) into Eq. (Cll) one can obtain Eq. (C12).

P =m XQ H

6021

31

9

-21

9

21

21

-34

6'l

LV

' . 2 'Derivation of the Coupled Stiffness Term (£2 mr x.)a

The virtual work due to this term can be formulated as

(C12)

P'P-2 'mr x. 6h dx

o (C13)

Assuming that the length of the finite element is small in comparison to r,Eqs. (Cl), (C7), and (C13) yield the following expression:

P =ft mr x9

12-

"

-6

a

6

-SL

-6

-a

6

I

91

K. (C14)

Determination of the Coupled Mass and Stiffness Matrices

Mass and stiffness matrices in bending and torsion can be combined toobtain a single mass-stiffness representation of the finite element fordescribing the coupled behavior. Combining the mass and stiffness matricesfrom Eqs. (C5), (C6), (C8), (CIO), (C12), and (C14), and defining theforce and displacement vectors for the element as

23

p = VP2

Tl

P3

P4

. V

£. = " p i~

P2

01

P3

P4

_ V

(CIS)

the coupled mass and stiffness matrices for the element can be written as

156

Symmetric

M.m£420

147xe 2l£xe 140(IQ/m+xe)

54 13£ 63xrt 156

-13A -3A -14Axe -22£ 4£

63x .70(I0/m+xe) 147xQ -21£xQ

(C16)

24

H to^O ^j

.CM

0 +•H

4-> W <=>?

C/D . F-I OH|o? H -I

1

h-l CNH - < W o ?W f) \D

t-t 1

CNCD

XH o?

CD+ X

CD in CNo? x B -i

CD ^H CM CNHH tO S C?

CM CMG CS i

f-< 00? LO ^ rHH .-H o? 0CN o? 1 H tO

CD+ X t-H CM 1

fn CM Wo?(-1 g ^H vD I-H .W o? CM Wo?

\o l^oHO?\O|LO p_ O CD I L_ O

^ ^H X •-!+ ^ CM H+ g w to +

l-t CM CN o?W t O I - H C N CS •-! 1— 'CMCM o? W o? W o?

o?CD

h~^

CMG

^/ — \

CMCD

XH

CJV '

o?

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g

CM

CMr-t

CDX

^

o?CD

CMCS

H

CMCD

X

3N '

0?

CDXf-t

CDX

CM

CN

o?

CMrH

CN

|

25

REFERENCES

1. Houbolt, J. C., and Brooks, G.W.: Differential Equations of Motionfor Combined Flapwise Bending, Chordwise Bending, and Torsion ofTwisted Non-Uniform Rotor Blades. NACA Note 3095, 1957.

2. Isakson, G., and Eisley, J. G.: Natural Frequencies in Coupled Bendingand Torsion of Twisted Rotating and Nonrotating Blades. NASA CR-65,1964.

3. Garland, C. F.: The Normal Modes of Vibrations of Beams Having Non-collinear Elastic and Mass Axes. J. of App. Mechs., Sept., 1940.

4. Perisho, C. H. : Analysis of the Stability of a Flexible Rotor Bladeat High Advance Ratio. J. of Am. Helicopter Soc., Vol. 13, No. 3,1968.

5. Przemieniecki, J. S. : Theory of Matrix Structural Analysis. McGraw-Hill Book Co., New York, 1968.

26

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Cu

rHCS0

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r*t

Cu

0)

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00

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oo <n0 ~.to •O 13

• CISt" rH

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r— 3

^f2

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OH

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j<0n

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to j-0 0CN tO

a>DO£•H

27

TABLE 2. Uncoupled Frequencies of CH-34 Blade in Bending and Torsion:

Mode No.

1234

Bending (rad/sec.)

25.272.4

136.2210.8

Torsion (rad/sec.)

173.6532.1934.9

1350.0

TABLE 3. The Frequencies Corresponding to the Coupled Modes, ConsistingMainly of First Mode in Bending and First Mode in Torsion.

(m)

0.00.020

.041

.050

.060

.069

.077

.083

.104

.124

.145

.187

2 2 2D (rad /sec. )D

635.0632.8626.2622.4616.3611.0604.9600.5581.9559.6539.0473.7

2 2 2u>T (rad /sec. )

30,13731,95938,49947,28459,34983,710

146,223289,632-71,923-31,572-18,779

-9,520

28

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31

-2 8 9 10 q

-20

Fig. 1 Variation of Coupled Frequencies of Rotating Blades Using:

s - Qq) Qq)1" (p-q) (P-1)

s8J

-2 -1

-1

-2 _

p = .8

X = 5.0

4,5 6 7 8 9 10 q

P'

II /

Fig. 2 Variation of Coupled Frequencies of Rotating Blades Using:

(1-q) (A-q)

32

max

max

1.0

.8

.6

.4

.2

0

-.2

-.4

-.6

-.8

-.8

-1.0

a) Bending Modes

- 1st Mode

- 2nd Mode

- 3rd Mode

1.0

O - 1st Mode

A - 2nd Mode

- 3rd Mode

b) Torsional Modes

Fig. 3 Uncoupled Natural Vibration Modes of CH-34 Blade,33

300 -

200 -

(cm2)

100-

O First Mode

Second Mode

0Frequency Ratio - q

Fig. 4 Variation of the Two Lowest Frequencies Correspondingto a Mainly Bending Mode with Increasing Eccentricity.

2 2, i xe (cm )

I- 200

- 100

-400 -200 200 400

Frequency Ratio - qFig. 5 Variation of the Lowest Frequency Corresponding to a

Mainly Torsional Mode with Increasing Eccentricity.

34

"NR,

40

302

20 .

10 .

1st Mode

2nd Mode

3rd Mode

10 . 15 25 30135

Fig. 6 Variation of Lowest Bending Frequencies for a Hinged Blade(CH-34) with Blade Rotational Velocity, (XQ = 0.0 and 3.28),

D

1st Mode

0).NR,

10

3-5

Fig. 7 Variation of Lowest Bending Frequencies, for a CantileverBlade (CH-34) with Blade Rotational Velocity.

35

U).B 120-(rad/sec)

100-

Fixed End— Hinged in Bending

1st Mode

35ft (rad/sec)

Fig. 8 Variation of Lowest Bending Frequencies for a Cantileverand Hinged Blade (CH-34) with Blade Rotational Velocity.

= 35

ft = 23.1

= 16.5

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0Fig. 9 Variation of the Torsional Rotations at the Blade Tip

with Eccentricity.

36

max ,6.

,1 .2

xr

Fig. 10 First Uncoupled Bending Mode for CH-34 and Fixed EndCH-34.

max

1.0,

.8

.6-

.4-

.2.

0

fl = 23.1

n = 0.083

-Fixed EndCH-34 CH-34

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

xr

Fig. 11 Torsional Component of First Coupled Mode for CH-34 andFixed End CH-34.

37

700 .,

600 .

500 .

400 .

300 .

200 .

100 .

XQ = 0.083

"" —6th Bending

Fig. 12 Interaction Between Mainly Torsional Modes and Higher BendingModes.

38

h_hmax

1.0

.8

.6

.4

Bending Component

1.30 W

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

xr

e ..max

1.0

.8 .

.6

.4

.2 ]

0

Torsional Component

W

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

xr

Fig. 13 Variation of First Coupled Modes with 30% Increase inWeight Distribution (W), (CH-34).

39

max

1.0

.8.

.6.

.4.

.2 .

0

-.2

i 1 1 1 1 1 1.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

x_r

Fig. 14 Variation of Second Coupled Mode with 30% Increasein Torsional Inertia (CH-34).

700,

600.

500

(rad/sec)

400.

300.

200.

100

x = 0.082

= 0.0

= 0.082

= 0.0

(rad/sec)

5 10 15 20 25 30 35Fig. 15 Variation of Torsional Frequencies with 30% Increase

in Torsional Inertia (CH-34).

1.0 -i Torsional Component

.8 -

0 .6 -

max JG1.30 JG

.1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

xr

Fig. 16 Variation of First Coupled Mode with 30% Increase inTorsional Rigidity (CH-34).

700 ,

600

500

400

300

200-

100-

0

x. = 0.082

(.rad/sec)

x_ = 0.082a

= 0.0

10 30 3515 20 25ft (rad/sec)

Fig. 17 Variation of Torsional Frequencies with 30% Increase inTorsional Rigidity (CH-34)

41

T2 6ll I 1C

Force

x (5 = x/A)

Displacement

Fig. 18 Nodal Forces and Displacements for a Blade Element inBending and Torsion.

42 NASA-Langley, 1974 CR-2379

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