ARMY RESEARCH LABORATORY

Vibration Analysis Applied tothe Motions of a Gas Turbine Engine

by T. A. Korjack

ARL-TR-1750 August 1998

A%;10

Approved for public release; distribution is unlimited.

The findings in this report are not to be construed as an officialDepartment of the Army position unless so designated by otherauthorized documents.

Citation of manufacturer's or trade names does not constitute anofficial endorsement or approval of the use thereof.

Destroy this report when it is no longer needed. Do not returnit to the originator.

Army Research LaboratoryAberdeen Proving Ground, MD 21005-5067

ARL-TR-1750 August 1998

Vibration Analysis Applied tothe Motions of a Gas Turbine Engine

T. A. KorjackInformation Science and Technology Directorate, ARL

Approved for public release; distribution is unlimited.

Abstract

A method has been suggested to compare and evaluate vibration environments or sinusoidaland random specifications through the use of a single analysis technique. The comparisonmethodology utilized in this analysis could be significantly implemented and exploited ininterpreting spectrum analysis results of vibration data such as in the analysis of a gas turbineengine of a modem tank. A generalized, multi-degree-of-freedom (DOF) lumped parameterstructural system model was used to allow an initial evaluation of the environments such asshock, sine, and random so as to possibly eliminate a detailed separate dynamic analysis for eachenvironment. Results indicate a possible reduction of analysis time and costs. In addition, sinceboth sine and random analyses depend heavily on modal damping assumptions for the accuracyof their predictions, the simple methodology proposed herein should prove to be useful andproductive.

ii

Table of Contents

Page

1. Introduction .................................................... 1

2. M ethodology ................................................... 1

3. C onclusions .................................................... 11

4. R eferences ..................................................... 13

Distribution List ................................................. 15

Report Documentation Page ....................................... 17

INTENimoNALLY LEFT BLANK.

iv

1. Introduction

Vibration analysis involving designs should be evaluated for a variety of loads in order to

determine the severity and possible failure. Not one environment (viz, shock, sinusoids, or random

excitation) allows the design of all structural components and, hence, we are led into separate

analyses for shock-random or periodic motions.

This work purports an approach, although quite simple, to compare and evaluate vibration

environment or sinusoidal and random specification via a single analysis technique. This

comparison methodology will prove to be extremely useful in interpreting spectrum analysis results

of vibration data that will be taken of the gas turbine engine of a typical modem tank, especially

pertaining to the turbine engine diagnostics (TED) project (Helfman, Dumer, and Hanratty 1995).

2. Methodology

In general, consider an il (multi) degree-of-freedom (DOF) lumped parameter structural system

with mass matrix [MJ], stiffness matrix [kj], damping matrix [Cij], and column matrix of external

forces {F (x., t)}. MK kj, Cij, and Fj are expressed in the w coordinate system. Given a forcing

function in the form as

{F (x., t)} = { P p (x.)} f (t), (1)

then, the differential equations of motion in the w coordinate system take the matrix form

[m]{W} + [c]w + [k]w = PI {p (x.)}f (t). (2)

If we apply a coordinate transformation,

1

{W} = [c] {'I}, (3)

in which each column of 4) is a modal column of the system and {rI} represents the normal

coordinates, then, from equation (3),

{-P} =[] {Wi} (4)

and

{WP} = [W]{1}. (5)

Substituting equations (3)-(5) into equation (2) and premultiplying by the transpose of [4)],

equation (2) becomes

[p)]T [in] [4p] {•I} + [1)]T [c] [4)] {rl} + [p)]T [k] [4)] {rl} = po [ )]T {p (xj)} f(t). (6)

From the orthogonality relations of natural modes, it follows that

[4)]T [M] [4)] = [M], (7)

and

[4)]T [k] [4)] = [ Mj] Mr] - K[1. (8)

Comparing the triple matrix product

[pf]T [c] [4)] (9)

2

with equations (7) and (8), this product will result in a diagonal matrix only when the damping

matrix [c] is proportional to either the mass matrix [m] or the stiffness matrix [k]; that is,

[c] =2PM, (10)

[c] = l[k], (11)

or

r= Cr wr, (12)

where C, is the ratio of the assumed to critical damping and Wr is the angular frequency. Using

equation (10) in equation (9) and comparing with equation (7),

[4ý]T [C] [4)] = 2P [Mr]. (13)

The substitution of equations (7) and (8) with equation (6) results in a set of n decoupled differential

equations of motion.

[Mr]{fi} + 2PN[M]{f1} + [6)r2][M] {I1} = [f pr{p(xj)} P. f(t). (14)

The rth equation of equation (14) has the form

M r + 2P3M lr + 2 MrTIr = Po{40(r)}{p(xj)}f(t). (15)

Let

{4(r)} {p (xj)} = = participation factor. (16)

Dividing through by MK in equation (15) and using equation (16) results in

3

Pi]r + 2P 'ir + 0r lr 0 f(t). (17)

r

A general form of solution to equation (17) is obtained readily by the use of Laplace transform

(Hurty and Rubenstein 1964; Gardner and Barnes 1942). For the case of shock spectrum, the

solution to equation (17) becomes

PPfir W o2r D (t), (18)

(A ~ 2 M rr r

in which

Dr(t) = f h (t - r) f(r) d'r

02 e-t-)=fot r

r P2

in [6 , p2 (t - t)] f(t) dt,r (19)

where Dr(t) is called the dynamic load factor.

The total deflection of the structure, w (x,t), is obtained by inserting equation (19) into

equation (3). Thus,

n

w (t)=PO D, (t). (20)W( 0i~ ~ M.

1 1

Starting again from equation (17), and posing the problem of harmonic or sine excitation,

f (t) = ei~t (21)

Use of Laplace transform provides the steady-state solution given by

4

Pr 1Po rr

f to _ _ _ _ f (t) (2 2 )fr~t -c2M 1 020

r + i2(rr r

o r H- () f (23)

r r

where

r =1,2, ... , n; H (Q) 1 (24)

1- + i 2Cr

\r r

The total deflection of the structure, w (xit), is obtained by inserting equation (22) into equation (3).

Thus,

n I"w(x,t) = PO 1 fr H. (Q) f(t), (25)

i~ 3Mi

or

nw(x,t)= P Z (k Trl (t) (26)

r=1

The mean square of the response can be written as

w2 (xt) = lrn - f w 2 (xt) dt. (27)T-- 2T -T

Substituting in equation (27) from equation (26) and then from equation (23), we have

___n n

w 2 (x,t) = lirn I• O (') 4~s (x) -, H T nr (t) rI (t) dtT-- r1 sIT

n n p21 r a r

E=1 a r 9 M Tlm I fT Hr (0) H (Q) f2 (t) dt. (28r=ls=l OO2a)M M T-- 2T -T(8r 9 r 1

5

If, as an approximation, we disregard phase relations that will tend to result in a higher mean square

value in equation (28) (i.e., neglecting terms that are a significant departure from the mean

[anomalies]), then the integral on the right-hand side of this equation can be written as

lia 1 ~T

lim - H (Q) H (Q) f 2 (t) dt. (29)T- 2-T

When the forcing function f(t) is a representative record of an ergodic process, we can transform

the limiting process of equation (29) from the time domain to the frequency domain because the

function f(t) is then represented by frequency components in a continuous frequency spectrum

0<f0<-o. Thus,

f 2(t) = ira 1 fT f 2 (t) dt - 1 fo f () dQ . (30)T-- 2T _T 2Tr

Using this transformation from the time to the frequency domain in equation (29), we write

1 1 f- (31)Tin-- J r (Q)1 H (•) f2 (t) dt f2 . -i (Q I-H (Qýf (0) dQ . (31)

Substituting from equation (28), we write for the mean square response of an n DOF system excited

by a random forcing function:

w G __ _2o 1 M fXIH (O)k (Q)If(0)dQ. (32)r-1 -i o~ z9M M 27

r t r t

The solution of equation (32) can be simplified further assume a lightly damped, multi-DOF system,

i.e.,

H ()= [(1 - + 4C 2] . (33)

6

According to Hurty and Rebinstein (1964), the magnification factors H (Q)l have regions of

pronounced peaks in the neighborhood of the corresponding natural frequencies Wr. Furthermore,

Biggs (1964) tells us that the products H (Q) and 1H (Q) for r # s are seen to be small in

comparison with the same products for r = s. In addition, in equation (32), terms with r * s may be

negative as well as positive, depending on the sign of the product qr 4b 1 F, while terms with4) ) r rG

r = s are always positive. Equation (32) will therefore be small. Hence, by disregarding the cross-

product terms corresponding to r = s, equation (32) becomes

n p2 r2 1 H 1w 2 (xt) = E 4)r22 (Q) 1 f (0) dQ". (34)

r ~r 1,) (34)nr r

Equation (34) can be further simplified by approximating the integrals of equation (34) by replacing

f (Q) by its discrete values f(6)r) at the natural frequencies wt and using

I_ f((r)d f(W) (o r f(co() r2 7c [( J)1 2 C2 02 27c 4 Cr (35)

1~ ~ -- + •

By introducing the following definitions

f(Wo = g' at f, (36a)

g' = g/t =g 2 /-Hz at f, (36b)m iputinput n

f = f = 2wrot, (37)

n r

and

1Q = - 2i(38)

7

equation (34) becomes

n 2 r 2 17~pp ( \)42 X

w2 (x,t) =r )2 (×) 0 - 2-gifQ" (39)r r o 4 M 2 ( 2 )

r r

Equation (39) is the equation that forms the basis of the method of response spectrum analysis.

Hence, in this perspective, we have restricted our analysis to the following constraints:

"* Ignored phase relations,

"* Ignored magnification products at different frequencies,

* Lightly damped all modes (frequencies), and

* Used discrete values of the input spectrum at the natural frequencies.

These conditions are realistic if we are dealing with a system that is grounded by base mounts, and

that has harmonic frequencies. At this point, we can summarize and rewrite equations (20), (25),

and (39) as follows:

- Case no. 1-shock (from equation [20]):

n Pw(x t)= 02 r (x (X) D,(t),r--1 °(0 2 M

r r

where r = i = 1, 2, ... , n; or from equation (25),

n PP•(xt)- Z o r 4ýr(x) D, (t),r=1 O) M

r

where

8

d 2D'1 (t)-= dt2 D(t)

Case no. 2-sine (from equation [39]):

w(xt)- = Z 2 r (x) Hr() )f(t),r )2 M

r r

where r =i =1, 2, ... ,n; or

PrW(x,t) c r] ° ••r (X) Q f (t),

r r7

where

f1 (t) =w d2

dt2

and

1Q-

2(r7

(from equation [33]), when Q = ~r.

* Case no. 3-random:

n 02 r 2

2 (x,t) y o r 4(x) r2gX fQ , (40)r=1 &4 M Qm )

r r

where, in all cases, * (xt) is defined as response acceleration. In case no. 3, response gRs

is defined as

S= wy2 (x,t) = [ 2 (x) 7[ f Q , (41)

r r

9

which includes all frequencies (w, fQ and modes 4ýr(X) of the structural system under

consideration. If we look at ggR per mode or frequency, we are led to

po r 7x 1/2

2-gn f Q (42)r r

as a first step toward the overall gvs computation. By comparing equation (41) to case no. 2 and,

in turn, to case no. 1, we see that the only difference in the response of a multi-DOF structure at a

natural frequency, w, are the forcing function terms; that is,

case no. 1--shock:

D'r (t), (43)

case no. 2-sine:

Q f' (t), (44)

and case no. 3-random:

Sg=f Q (45)

All other terms of the aforementioned cases; that is,

P ro r (4r (X), (46)

0 2 Mr r

define a multi-DOF system (as opposed to a one-DOF system) and are common to all gre•¢

solutions for shock, sine, or random environments.

10

Equations (42)-(44) are nothing more than the grpoe terms as discussed for a simple one-DOF

system by Hurty and Rubinstein (1964).

Using the analysis technique for response spectrum analysis and saving the g, terms at each

natural frequency will then allow the determination of true g, response for each environment

(shock, sine, or random) that differs by a g&p magnitude difference defined and known via

equations (42)-(44).

3. Conclusions

The approach shown previously provides the benefits of time and cost savings but is limited by

the assumption required to derive the approach. The application of the approach can be verified

early in the design but requires the capability to solve the dynamic analysis using a program that

utilizes both approaches formulated by equation (28) vs. equation (39).

Notwithstanding the fact that shock response is little affected by assumed damping, both sine

and random analyses depend heavily on modal damping assumptions for the accuracy of their

predictions. Although the approach in this analysis was based upon the concept of response

spectrum analysis, its concept is used quite extensively in oscillatory systems and even in earthquake

analysis. Here, the simplified equations can be directly correlated to sine or shock vibration

analyses. Hence, by means of this simplified, but yet practicable technique, one can easily solve all

or any one of the designing environments for a given structure, such as the case for the mechanical

structural components/mechanisms inherent in a typical gas turbine engine assembly network.

However, this approach is feasible provided that phase relations are ignored and magnification

factors are not a problem. Furthermore, this methodology is especially useful for systems involving

harmonic frequencies based upon the natural frequencies of the individual mechanical components.

11

INTENTIONALLY LEFT BLANK.

12

4. References

Biggs, J. M. "Introduction to Structural Dynamics. " McGraw-Hill, 1964.

Gardner, M. F., and J. L. Barnes. "Transients in Linear Systems. " John Wiley Sons, Inc.,New York, NY, 1942.

Helfman, R., J. Dumer, and T. Hanratty. "Turbine Engine Diagnostics." ARL-TR-856, U.S. ArmyResearch Laboratory, Aberdeen Proving Ground, MD, September 1995.

Hurty, W. C., and M. F. Rubinstein. "Dynamics of Structures. " Prentice-Hall, Inc., EnglewoodCliffs, NJ, 1964.

Thomson, W. T. Laplace Transforms (2nd ed.). Prentice-Hall, Inc., Englewood Cliffs, NJ, 1960.

13

INTENTIONALLY LEFT BLANK

14

NO. OF NO. OFCOPIES ORGANIZATION COPIES ORGANIZATION

2 DEFENSE TECHNICAL 3 DARPAINFORMATION CENTER L STOT'SDTIC DDA J PENNELLA8725 JOHN J KINGMAN RD B KASPARSTE 0944 3701 N FAIRFAX DRFT BELVOIR VA 22060-6218 ARLINGTON VA 22203-1714

HQDA 1 US MILITARY ACADEMYDAMO FDQ MATH SCI CTR OF EXCELLENCEDENNIS SCHMIDT DEPT OF MATHEMATICAL SCI400 ARMY PENTAGON MDN A MAJ DON ENGENWASHINGTON DC 20310-0460 THAYER HALL

WEST POINT NY 10996-1786OSDOUSD(A&T)/ODDDR&E(R) 1 DIRECTORR J TREW US ARMY RESEARCH LABTHE PENTAGON AMSRL CS AL TPWASHINGTON DC 20301-7100 2800 POWDER MILL RD

ADELPHI MD 20783-1145CECOMSP & TRRSTRL COMMCTN DIV 1 DIRECTORAMSEL RD ST MC M US ARMY RESEARCH LABH SOICHER AMSRL CS AL TAFT MONMOUTH NJ 07703-5203 2800 POWDER MILL RD

ADELPHI MD 20783-1145PRIN DPTY FOR TCHNLGY HQUS ARMY MATCOM 3 DIRECTORAMCDCG T US ARMY RESEARCH LABM FISETTE AMSRL CI LL5001 EISENHOWER AVE 2800 POWDER MILL RDALEXANDRIA VA 22333-0001 ADELPHI MD 20783-1145

DPTY CG FOR RDE HQ ABERDEEN PROVING GROUNDUS ARMY MATCOMAMCRD 4 DIR USARLBG BEAUCHAMP AMSRL CI LP (305)5001 EISENHOWER AVEALEXANDRIA VA 22333-0001

INST FOR ADVNCD TCHNLGYTHE UNIV OF TEXAS AT AUSTINPO BOX 202797AUSTIN TX 78720-2797

GPS JOINT PROG OFC DIRCOL J CLAY2435 VELA WAY STE 1613LOS ANGELES AFB CA 90245-5500

15

NO. OFCOPIES ORGANIZATION

1 DIR ARLAMSRL IS J GANTT2800 POWDER MILL RDADELPHI MD 20783-1197

1 DIR ARLAMSRL IS C M KINDLGIT 115 O'KEEFE BLDGATLANTA GA 30332-0800

ABERDEEN PROVING GROUND

7 DIR, ARLAMSRL IS C,

J DUMERR HELFMANR KASTET KORJACK (4 CP)

16

I Form ApprovedREPORT DOCUMENTATION PAGE oMp No. 0704188

Public reporting burden for this collection of Information It estimated to average 1 hour per response, including the tit.e for reviewing Instructions, serchling existing data scurcea,gathering and maintaining the data needed, and completing and reviewing the collection of Information. Send comments regerding this burden eatimote or any other aspect of thiscollection f Information, including suggestions for reducing this burden, to Washington Heedqatrers Services, Directorate for Information Operations end Reports, 1215 JeffersonDavis Hiohway. Suite 12'04. A rllnofon. VA 2 2203-4302. and to the Office of Manenemlent and gudnel. Panerwork Reduction PyoiectfO7OA-O188r Weshinoton. DC 20503.

1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

I August 1998 Final, October 1997 - February 19984. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Vibration Analysis Applied to the Motions of a Gas Turbine Engine

4B0105033500006. AUTHOR(S)

T. A. Korjack

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATIONREPORT NUMBER

U.S. Army Research LaboratoryATTN: AMSRL-IS-CI ARL-TR-1750Aberdeen Proving Ground, MD 21005-5067

9. SPONSORINGJMONITORING AGENCY NAMES(S) AND ADDRESS(ES) 10.SPONSORING/MONITORINGAGENCY REPORT NUMBER

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE

Approved for public release; distribution unlimited.

13. ABSTRACT(Maximum 200 words)

A method has been suggested to compare and evaluate vibration environments or sinusoidal and randomspecifications through the use of a single analysis technique. The comparison methodology utilized in this analysiscould be significantly implemented and exploited in interpreting spectrum analysis results of vibration data such as inthe analysis of a gas turbine engine of a modem tank. A generalized, multi-degree-of-freedom (DOF) lumped parameterstructural system model was used to allow an initial evaluation of the environments such as shock, sine, and random soas to possibly eliminate a detailed separate dynamic analysis for each environment. Results indicate a possible reductionof analysis time and costs. In addition, since both sine and random analyses depend heavily on modal dampingassumptions for the accuracy of their predictions, the simple methodology proposed herein should prove to be useful andproductive.

14. SUBJECT TERMS 15. NUMBER OF PAGES

18vibration, structures, spectrum analysis 16. PRICE CODE

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS PAGE OF ABSTRACT

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED ULNSN 754121-280-5500 Standard Form 298 (Rev. 2-89)

17 Prescdbed by ANSI Std. 239-18 298-102

INTENTIoNALLY LEFT BLANK

18

USER EVALUATION SHEET/CHANGE OF ADDRESS

This Laboratory undertakes a continuing effort to improve the quality of the reports it publishes. Your comments/answersto the items/questions below will aid us in our efforts.

1. ARL Report Number/Author ARL-TR-1750 (Koriack) Date of Report August 1998

2. Date Report Received

3. Does this report satisfy a need? (Comment on purpose, related project, or other area of interest for which the report will

be used.)

4. Specifically, how is the report being used? (Information source, design data, procedure, source of ideas, etc.)

5. Has the information in this report led to any quantitative savings as far as man-hours or dollars saved, operating costsavoided, or efficiencies achieved, etc? If so, please elaborate.

6. General Comments. What do you think should be changed to improve future reports? (Indicate changes to organization,

technical content, format, etc.)

Organization

CURRENT Name E-mail NameADDRESS

Street or P.O. Box No.

City, State, Zip Code

7. If indicating a Change of Address or Address Correction, please provide the Current or Correct address above and the Old

or Incorrect address below.

Organization

OLD NameADDRESS

Street or P.O. Box No.

City, State, Zip Code

(Remove this sheet, fold as indicated, tape closed, and mail.)(DO NOT STAPLE)

DEPARTMENT OF THE ARMYNOPOSTAGE

NECESSARYIFMAILED

OFC•AL BUSINESS IN THEUNITED STATES

BUSINESS REPLY MAILFIRST CLASS PERMIT NO 0001,APG,MD

POSTAGE WILL BE PAID BY ADDRESSEE

DIRECTORUS ARMY RESEARCH LABORATORYATTN AMSRL IS CIABERDEEN PROVING GROUND MD 21005-5067