VOLUME 9, NQO. 2

FEBRUARY 197 7

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THE SHOCK111IBRATION'DIGEST 4L

A PUJBLICATION OFTHE SHOCK AND VIBRATIONINFORMATION CENTERNAVAL RESEARCH LABORATORYWASHINGTON, D. C.

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THE SHOCK AND VIBRATION DIGEST

Volume 9 No. 2February 1977

STAFF

EDITORIAL ADVISORS: Henry C. PuseyRobert Belsheim

TECHNICAL EDITOR: Ronqld L. Eshleman

EDITOR: Judith NagleEshloman

RESEARCH EDITOR: Milda Tamulionis

BOARD OF EDITORS: R. Belsheim W. D, PilkeyR. L. Bort A. SemmelinkJ, 0. C. Crisp E. SevinC. L. Dym J. G. Showalterb. J. Johns R:,A. SkopG. H. Klein C. B. SmithK. E. McKee JrC.SnowdonJ. A. Macinante R. H. VolinC. T. Morrow H. von GierkoJ. T. Oden E, E. Ungar

of The Shock and Vibrtion'Digest, is a monthly publication of the Shock andVibration Information Center. The goal of the Digest is to provide efficient

THE SHOCK AND VIBRATION transfer of sound, sock, and vibration technology among researchers and

INFORMATION CENTER practicing engineers, Subjective and objective analyses of the literature are pro,vidd along with news and editorial material. News items and articles to be

Code 8404 Naval Research Laboratory considered for publication should be submitted to:

Washington, D.C., 20375

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Director Vibration InstituteSuite 206

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J. Gordan Showalter

Copies of Articles abstracted are not available from the Shock and VibratiorRobert Belsheim Information Center (except for those generated by SVIC). Inquiries should be

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DIRECTOR NOTES

Looking back at previous Shock and Vibration symposia, a matter of perpetualconcern has been the effectiveness of oral presentations and, in particular, thequality of visual aids used by the speakers. This problem is not unique to oursymposia; it is recognized by every technical society concerned with the organiza-tion of technical meetings. The conference management in each case persistentlyprepares guidelines which are faithfully delivered to each accepted speaker, yetsome poor presentations are still with us. The problem is recognized by managers ofresearch institutions. Dr. Alan Berman, Director of Research at the Naval ResearchLaboratory recently insisted that NRL briefings be improved by eliminating illeg-ible and cluttered slides. It is highly probable that most speakers are aware of theproblem, yet many still violate the basic rules for planning their presentations andpreparation of their visual aids. In doing this they are performing a serious dis-service to themselves and to their prospective audience,

I do not propose to give detailed guidelines here, but I can recommend an excellentarticle by Hubbard* for those who are interested in improving their presentations.In general, speakers should remember that their audience is not already familiarwith their work. They should avoid complex mathematical derivations and shouldkeep their slides simple and descriptive. It is distressing when researchers withsignificant accomplishments fall down in their efforts to communicate to others.

The 48th Shock and Vibration Symposium is now firmly scheduled for 18-20October 1977 in Huntsville, Alabama. The U.S. Army Missile Command will behost. The first call for papers will be issued next month. It is hoped that all pro-spective speakers will do their homework on effective presentations. This effort willhelp to make the 48th Shock and Vibration Symposium a useful experience for all.

H.C.P.

•H.H. Hubbard, "Guidelines for the Planning and Preparation of Illustrated Techni-cal Talks," J Acoust. Soc Amer., QQ (5) (Nov 1976)

nIEDITORS RATTLE SPACE

MACHINE DIAGNOSIS AND PROGNOSIS: A NEW TECHNOLOGY

Machine diagnosis and prognosis is a new formal technology. The availability ofcomplex signal processing equipment and sensors has enabled the engineer to usenatural vibration and acoustic responses to carry out formal machine diagnosis. Thisnew technology allows engi,;oers to determine shut-down times for reasons ofsafety and to avoid loss of productivity. No longer does a machine have to run to

destruct ion.

What is involved in machine diagnosis? First, the engineer must know the physicsof the machine and its functions. Second, he must be able to measure and analyzemachine vibration signals. Finally, he must be able to interpret the data properly.

Engineers tend to try to diagnose machine faults without knowing the physics ofthe machine, particularly its vibrational properties. Yet, how can the engineerassociate vibration signals with machine faults or conclude that a problem is one ofwear, misuse, corrosion, or process malfunction unless he knows about resonances,critical speeds, and conditions of stability? It is not sufficient to draw parallels

between the frequencies of vibration signals and malfunction frequencies.

Many engineers try to match vibration signals with an excitation such as massunbalance while watching for an increase in vibration level. Such a technique mayoccasionally give valid results, more often, however, the problem is too complicatedto be so simply resolved. In order to solve machine fault problems routinely, theengineer must know his machine and why it vibrates at given levels.

In the near future machine prognosis will most likely become practical. Machineprognosis will allow the engineer to determine the most advantageous time forshut down. It will thus be possible to schedule repairs with no loss of production.The engineer will probably make a series of vibration measurements over a periodof time. These measurements will be processed by a mathematical model of themachine and its process, In this way the state of the machine at some set date inthe future can be determined. An alternative would be to determine when themachine will fail, so that maintenance can be scheduled. Prognosis is not yetpractical because suitable wear models are not yet available and changes in machineexctation and conditions affecting stability as the machine is used cannot bepredicted Thus machine vibration is on the way to becoming an asset rather than .

liability

R.L.E.

2

ABSORBERS AND ISOLATORS FOR TORSIONAL VIBRATION

John M. M. Vaee

ThEis decl3 deie t for reducing torsonalvibrtion in rotating machinery. Conmiment definitions armgiven for absorbers, isolators, and dampomr

Devices for reducing torsional vibration in rotating kFmachinery can be classified as absorbers, isolators,or dampers. These terms have been used inconsis-tently in the literature, and the definitions have been T sin Wtinterchanged. To further confuse the issue, somedevices are actually a combination of two, or eventhree, of these classes. The dfinitions used in thisarticle are consistent with the classical literature onvibration theory. It is hoped that this will help the kreader understand how a given device is meant to Llqlwork. Ti' ' 8

The number of devices for reducing torsional vibra- 6tion that have been successfully applied is smallcompared to the number of ideas published andpatents granted. The number of devices now being 2sucessfully marketed is also small, despite a demand _

that existing manufacturers cannot meet. Further- 0 1: 0 , Imore, most of the published literature on the reduc- Wn

tion of torsional vibration is old; relatively little new (b) Response of e1information has appeared in the last decade.

Figure 1: Torsional Absorber and ResponseBecause this is the first of a series of reviews on the of Main Inertia with Absorber Attachedreduction of torsional vibration, older literaturewhich the author believes is informative or poten-tially useful is briefly described Practical problems associated with these absorbers are

that they function only at one frequency, that theAosorbers amplitude of 92 becomes so large as to break k2 , orTorsional vibration absorbers are based on Frahm's that there is not enough space or support strengthprinciple: an inertia 12 is attached to the vibrating for the required size of 12. Much of the literature onmember with a spring k2 that is tuned to be in torsional absorbers has been concerned with novelresonance with the offending frequency. The added designs aimea at overcoming these problems.inertia appears infinite at the tuned frequency andconverts the vibrating point to a node [11. Figure 1 An early design was the pendulum damper*. Theillustrates the basic idea for the case in which the inertia is a mass pinned and rotating so that theexciting frequency wi in near the natural frequency pendulum is aligned with a radius of the shaft 12].of the main system. This is the case for which an The centrifugal force field produces a natural fre-absorber will most likely be needed. quency that is proportional to shaft speed, thus

*Department of Mechanical Engineering, University *Commonly used terminology, but Incorrect.of Florida, Gainesville, Florida 32611

3

eliminating the single-frequency characteristic of the system far to the right on the response curve forsimple absorber. One of the few recent successful 12 (large W/Wa, Wa = k2/I2).

efforts to apply the pendulum damper involved ahelicopter rotor [3]. Vibration of such rotors in- Torsional isolators for rotating machinery take thecludes both translational and torsional modes. form of shaft couplings with low torsional stiffness.

A coupling with elastic elements and both lowAn absorber has been described (41 in which the stiffness and high torque transmission capabilityspeed-proportional natural frequency is obtained by presents a difficult design problem; therefore, mostusing an electromagnetic torsional spring; the stiffness elastic isolator couplings are hybrids that incorporatevaries with field current, Figure 2 illustrates the damping with moderately low stiffness.mechanism of another absorber (51. The reaction ofthe inertia is through Coriolis forces, and the device Very low torsional stiffness has been combined withdeparts somewhat from the basic Frahm principle, high torque capability by designing couplings withIt is shown to be more effective than a pendulum nonlinear stiffness characteristics. These couplings usedamper when the order" of torsional vibration is the centrifugal force on connecting links (betweenless than two. flanges) to produce torsional stiffness that is depen-

dent in a nonlinear manner on speed and transmittedtorque. Such, a design [6] has been successfully

To $in Wt tested in marine drives (to protect the gearbox intugboats). A similar design has been developed inRussia [71.

It is possible to design centrifugal link couplingsMhaving zero torsional stiffness. This seeming anomaly

has been explained [8], as has an application of thecouplings to helicopter drive trains [8-10] . Althoughzero stiffness may be undesirable, (due to a 'hunting'

I characteristic), a small enough positive stiffness canpractically eliminate the two-node resonance in athree-inertia drive train (6].

Dampers

A torsional damper is defined in this article as anydevice designed to dissipate the kinetic energy oftorsional vibration into heat. Torsional dampers can

be subdivided into two types: devices in which theFigure 2: Centrifugal Governor Mechanism damping eiement transmits none of the useful torque;

Used as a Torsional Absorber and devices in which the element transmitting thetorque also contains the damping. The second typealways utilizes a specified and controlled stiffness

Isolators in addition to the damping; it is discussed in the next

Suppose that 12 in Figure 1(a) is not an inertia added section.

to make an absorber, but is instead a rotating ma-chine component that is susceptible to malfunction Dampers in which no useful torque is transmittedorhdamge yoron al v that case susceptible to ucaon utilize an added inertia (flywheel). This inertia isor damage by torsho rional vibatIn of c by driven by the rotating shaft only through the forcebe iso lated fro m the to rsio nal vibratio n o f 11 b y o i h r c u o b ( r ) o i c u r c i n hmaking the torsional spring k2 as soft as practical. of either coulomb (dry) or viscous friction. The

added inertia can follow the speed oscillations only if

Successful use of an isolator depends on ensuring that the friction force is large enough to produce the

all natural frequencies are well below the excitation required accelerations, hence, slipping occurs under

frequency, as the basic principle is to operate the some conditions, thereby dissipating energy as desired

"Order is the number of cycles per shatt revolution.

4

The Lanchester damper was the original dry friction limit, irrespective of the frequency of the disturbingtype. The theory and experimental results have been torque." [lb).

described (11]. The disadvantage of dry friction isthat the friction coefficient is difficult to control; it Suggested Future Workvaries with temperature and is sensitive to oils and There is a need for experiments and publication ofother contaminants. Equations for the design of results for various types of devices that are eitherfrictional dampers, based on experimental measure- in use or appear promising. The published test dataments have been derived [12). available are old and practically obsolete for applica-

tion to modern systems.The viscous damper does not have the disadvantageof temperature and contaminant sensitivity, if a fluid Testing should involve the devices themselves andhaving a viscosity relatively insensitive to temperature their effects. , iffness, damping, and inertia proper-is used. A viscous damper in which silicone is the ties of the devices under operational dynamic condi-working fluid 1 13] is available commercially. The tions should be determined. The effects of devices ontheory and an analysis of a diesel drive application system response should also be studied. Such databased on considerations of energy dissipated per could allow the engineer to better apply these devicescycle have been presented (141. It appears to the where they are needed.author that the classical Holzer analysis (151 couldbe modified and used to calculate complex eigen.values (including the real part) so as to allow com- REFERENCESputation of the damping effectiveness of dampers.

1. Den Hartog, J.P., Mechanical Vibrations,Hybrid Devices McGraw-Hill, pp 87-93 (1956).Several devices available commercially employ acombination of the principles described above. All 2. Ibid., pp 219-222.of these torsional couplings employ some type of

elastic element .- i.e., radial leaf spring, rubber in 3. Paul, W.F., "Main Rotor Absorber Reducescompression, rubber in shear -- together with a Helicopter Vibration to 0.1 G," Space/Aero-mechanism for energy dissipation such as material nautics, 53, pp 46-48 (Mar 1970).hysteresis or fluid viscosity. Unless the excitationfrequency is high, as would be the case for gear tooth 4. Ching-U, I. and Morse, I.E., "Variable Speedimpact, elastic stiffness elements are not soft enough Torsional Vibration Absorber," Machine Design,to act as a true isolator. A change of stiffness is often pp 93-95 (Aug 7, 1958).sufficient to move the system off resonance and curea problem, however. So far as the author can deter- 5. Pringle, O.A., "Use of the Centrifugal Governormine, technical information on these devices is Mechanism as a Torsional Vibration Absorber,"obtainable only from commercial literature or pro- ASME Trans., 75 (1), pp 59-62 (Jan 1953).prietary reports

6. Chapman, C.W., "Zero (or Low) TorsionalOne complex hybrid device that has apparently not Stiffness Couplings," J Mech. Engr. Sci.,been marketed is the tuned and damped gyrostatic 11(1) (i969)vibration absorber (161. It is a combination ofabsorber and damper. A gyro wheel with the axis 7. Bal/hi, M F. and Esin, G D., "Flexible Metallic

mounted normal to the ma',,ne shaft is the added Couplings wiih Dynamic Links," Eng. Digest

inertia. The jyro speed provides 3 variable not possi- (Apr 1960).

ble with a pure absorber; the physical siue of the 8 Vance, J.M., Brown, R.A , and Darlow, MS.inertia thus is not such a limiting factor as in the pure "Feasibility Investigation of Zero Torsionalabsorber. The gimbal is restrained with springs and "dampers. "By a judicious choice of spring stiffness Stiffness Couplings for Suppression of Reso-and iscus ampng t , po~~bl toensre hatthenance and Instability in Helicopter Driveand viscous damping it is possible to ensure that the Trains," USAAMRDL.-TR-73-103, U.S. Armyamplitude -.-- will not exceed some predetermined Air Mobility Res Deveiopm. Lab Fort Eustis,

VA.

5

9. Vance, J.M. and Brown, R.A., "Suppressionof Torsional Vibration with Zero TorsionalStiffness Couplinos," U.S. Naval Res. Lab.,Shock Vib. Bull., 44, pp 43-54 (Aug 1974).

10. Darlow, M.S. and Vance, J.M., "TorsionalStability Analysis of a Gas Turbine PoweredHelicopter Drive System," J. Engr. Power,Trans. ASME, pp 335-341 (Oct 1974).

11. Den Hartog, J.P. and Ormondroyd, P., "Tor-sional Vibration Dampers," ASME Trans.,52, APM-52-13-133 (1930).

12. Chichinadze, A.V. and Ternish, O.S., "Design-ing a Torsion-Vibration Damper," RussianEngr. J., (1), pp 13-16 (Jan 1971).

13. Geschelin, J., "Welding Facilitates Assembly ofSilicone Engine Dampers," Automot. Indust.,97 (11), pp 24-25 (Dec 1,1947).

14. Den Hartog, J P, Mechanical Vibrations,McGraw-Hill, pp 210-216 (1956).

15. Ibid., pp 187-194.

16 Arnold, R.N., "Tuned and Damped GyrostaticVibration Absorber," lnstn Mech. Engr. Proc.,157 (25), pp 1-15 (1947).

i6

Asurvey and analysisLITERATURE REVIEWThe monthly Literature Review, a subjective critique and summary of the litera-ture, consists of two to four review articles each month, 3,000 to 4,000 words inlength. The purpose of this section is to present a "digest" of literature over aperiod of three years. Planned by the Technical Editor, this section provides theDIGEST reader with up-to-date insights into current technology in more than150 topic areas. Review articles include technical information from articles, reports,and unpublished proceedings. Each article also contains a minor tutorial of thetechnical area under discussion, a survey and evaluation of the new literature, andrecommendations. Review articles are written by experts in the shock and vibrationfield.

This issue of the DIGEST contains review articles on Dynamic Stiffness and Damp-ing of Fiber-Reinforced Composite Materials and Finite Element Modeling.

Professors Gibson and Plunkett of Iowa State University and University of Min-nesota respectively, review the recent experimental and analytical efforts to charac-terize the dynamic mechanical properties of fiber-reinforced composite materials.

Professor Krishna Murty of the Indian Institute of Science, Bangalore, reviewsfinite element modeling techniques including lumped parameter, transfer matrix,finite element displacement and finite element force. Hybrid and quadratic eigen-value methods are reviewed.

7

DYNAMIC STIFFNESS AND DAMPING OF FIBER-REINFORCED COMPOSITE MATERIALS

R.F. Gibson* and R. Ilunkett"

Abstract - This paper reviews rcent experknental about the fiber axes. Cross-ply laminates of unidirec-end anaytical efforts to characterize the dynmnic tional plies are often used when strength and stiff-mechanksi propertes of fiber-reinforced composite ness are needed in more than one direction.materiw

THEORETICAL CONSIDERATIONSMethods that utilize microscopic mechanics havebeen reported for finding the effective macroscopic Elastic Analysismoduli of fiber-reinforced composites, or ortho- Much of the work in micromechanics has been basedtropic materials (1 - 61 . Determination of macroscop- on static loading and deformation fields havingic dynamic orthotropic moduli using methods that characteristic lengths much greater than those of thecombine microscopic analysis and geometric informa- material. The so-called effective modulus theoriestion with the isotropic moduli of the constituent [131 used in this work are also applicable to dynami-materials are not as well verified. The purpose of this cally loaded composites when the vibrational wave-paper is to provide a crit'cal review of the literature length is much greater than the largest characteristicon the rapidly expaneing subject of composite dimension of the representative volume element.materials. The emphasis is on measurement; repre- That is, the averaging on which the effective modulussentative analytical approaches are mentioned, how- theories are based is valid so long as the deformationever, because they are referred to in discussions of gradients are sufficiently small. Continuum theoriesexperimental results. Most of the references are have been developed in order to study the propaga-concerned with dynamic properties of materials tion and dispersion of high-frequency harmonic wavesduring sinusoidal vibration. A few publications in composites 17 - 101 . References on wave propaga-dealing with the propagation and attenuation of tion are also given in the articles by Ross and Siera-elastic waves are included because the authors' kowski [111, Moon (121, and Achenbach 113].research interests have to do primarily with vibra-tions. With the exception of some analytical work The discussion that followj is primarily concernedon nonlinear effects and some experimental work with vibrations of sufficiently long wavelength thaton large amplitude vibration and the material damage effective modulus theories are applicable. With theseit causes, the results described are for strains below theories, dynamic properties differ from static pro-the elastic limits of the constituent materials. perties only when the anelastic iature of the matrix

(and to a much lesser extent, of the fibers) is takenThe material configurations of interest in this paper into account.are based on the unidirectional ply. Such a ply iscomposed of a number of parallel, continuous, rein- Various approaches to elastic micromechanical anal-forcing fibers embedded in a thin layer of binder ysis have been reported in the literature [14]. Hetero-(matrix) material. Reinforcing fibers are generally geneous materials in general have been reviewed 1151.composed of glass, boron, or carbon; the most com- The mechanics-of-materials approach [3, 41 involvesmonly used matrix materials are cured epoxy resins certain simplifying assumptions about the stress orexcept for high temperature applications. Unless strain distribution in a composite material; forotherwise stated, it is assumed that the unidirectional example, that plane sections of a composite beamply is orthotropic and statistically transversely iso- remain plane in pure bending. The longitudinaltropic, that is, the ply has two orthogonal planes strain is therefore linearly proportional to the dis-of material property symmetry, and the material tance from the centroidal axis. The exact stress orproperties are statistically invariant to rotations strain distribution in a composite material must be

*Assistant Pro.essor, Dept. of Engrg. Science and Mech., and Engrg Research Institute, Iowa State University,,Ames, Iowa 50011Professor, Dept of Aerospace Engrg. and Mech., University of Minnesota, Minneapolis, Minnesota 55455

9

-..-- - - . .. - .~-

known before its properties can be determined by It was concluded that the random array results arethe theory of elasticity. Alternatives to the solution preferred because the rardom array bounds coincide,of this formidable problem involve semi-inverse thus providing "exact" expressions for the moduli.techniques: part of the solution is assumed and then Randomness was introduced in the fiber array withchecked to determine whether or not equilibrium composite cylinders of various sizes containingand boundary conditions are satisfied. Variational fibers surrounded by outer cylinders of matrix mater-principles such as minimum potential energy, comple- ial. The remaining volume within the compositementary energy, or a mixed principle, can also be cylinder assemblage was filled by composite cylindersused to obtain alternatives to the exact solution, of smaller and smaller size. The same fraction of theMethods based on variational principles provide only total volume was maintained for each cylinder.bounds on composite properties, but, in some cases, As a result of this randomness and filling, the boundsthese bounds are very close to each other, and are converged in the limit. F-,r, a practical standpoint,sometimes coincidental, the introduction of randomness in the fiber array

model is more appealing than the imposition of aThe variational principles of minimum potential regular pattern of fiber packing that never occurs inenergy and complementary energy have been used to practice. Support for the random array hypothesisobtain coarse bounds on the elastic moduli of an has been provided by Adams and Tsai [21], whoisotropic, two-phase material of arbitrary phase developed finite element models representing randomgeometry (16). A unique mixed variational principle arrays, They found that the agreement between mea-that reduces to the principles of minimum potential sured and predicted moduli improves as the degreeenergy and complementary energy for two limiting of randomness in the fiber array increased.cases has been developed [171. The mixed principleinvolves comparison of two hypothetical materials The difficulty of finding exact solutions using theand their respective states of stress and strain. One theory of elasticity has stimulated the developmentmaterial is hqmogeneous, elastic, and isotropic, the of various numerical techniques [22). An interestingother is nonhomogeneous, elastic, and anisotropic method is the interpolation technique [23); it In-The extremum condition for the variational function- volves interpolation between previously derivedal is equivalent to the condition that the stresses and bounds on elastic moduli, The interpolation factorsstrains in the anisotropic material satisfy the proper are measures of the degree of reinforcement by theconstitutive equations. fibers and are chosen by comparison to other 'exact'

numerical solutions.This variational principle has been applied to ortho- The theories described thus far are based on lineartropic, transversely isotropic fiber-reinforced mater- elastic behavior of the composite material and itsials (18) .Bounds for three of the five independent components. Nonlinear elastic behavior may bemoduli were obtained in terms of phae moduli and caused by large displacements (geometric nonlinear-volume fractions. Hill (19] used a different method ity) or by matrix-governed nonlinearity. Nonlinearityto derive bounds on one of these moduli and on the of fiber-reinforced composites is more apparent intwo moduli for which bounds had not been obtained. thJr shear stress-shear strain response than in theirThe combined work thus provides bounds on all corresponding uniaxial stress-strain response [3].five moduli. The results were identical for the one Constitutive equations that can describe this nonlin-modulus bounded by both techniques It was claimed earity in shear have been formulated [24, 25]. Ac-that these are the best possible bounds obtainable cording to Timoshenko's theory for vibrating beams,

without taking phase geometry into account Closer shear deformation becomes more important as thebounds require additional constraints in the form slenderness ratio decreases, as the mode numberof specified fiber arrays Such bounds have been ob- increases, and as the ratio of the extensional modulustained for fiber-reinforced materials with hexagonal to the shear modulus increases. It has been shownand random ,.r arrays [20), both arrays are trans- analytically [26] that the effects of this nonlinearityversely isotropic on free vibration frequencies of undamped, laminated

beams become more important as the beam slender-ness ratio decreases and as the mode number in-creases

10

Anelastic Analysis wavelength is long in comparison with the charac-

Under certain conditions the mathematical models teristic dimension of the composite microstructure.)for the linear micromechanical theories discussedabove can be modified to include the dynamic The loss factor, or loss tangent, is the ratio of thebehavior of anelastic materials. Anelasticity is the imaginary part of the complex modulus to the realterm generally used to describe the stress and time part (271. The loss factor may also be defined as thedependence of elastic strain. That is, the strain lags ratio of the energy dissipated per cycle to the strainthe stress and the resulting hysteresis causes damping, energy stored in the material at peak displacement.or anelastic energy loss during vibration. A number of The effective loss factor for the composite may bemathematical models for material damping have been found from the elastic stiffnesses and loss factors ofdiscussed (27, 28, 291. Th:. complex modulus nota- the constituent materials and the cc.srespondingtion, which consists of a real part representing elastic volume fractions [35, 371.stiffness and an imaginary part representing dissi-pation, is widely used to model the behavior of linearanelastic materials under sinusoidal vibration [27, 28, MEASUREMENT OF DYNAMIC PROPERTIES301. It has been shown that the complex modulus ofa linear anelastic material may depend on vibration Experimental Techniquesfrequency, but not on amplittude [27]. With the Attempts to compare predicted results and actualelastic-viscoelastic correspondence principle (311, measurements of composite materials invariablyintegral transforms in the time domain are used to require the use of such mechanical properties of sim-obtain viscoelastic solutions from elastic ones. pIe beams, rods, and plates as stiffness, frequency,

dispersion, and damping ratio. Mechanical propertiesAlthough complex modulus notation was developed are used because it is difficult to determine the be-for application to homogeneous, isotropic materials, havior of the individual components except by photo-it may also be applied to composite materials having elastic methods and because many of the analyticalone or more anelastic phases. A correspondence methods neither use nor predict the details of theprinciple has been developed for isotropic viscoelastic stress and strain fields.

composites and applied to a particulate compositeI . 321. The elastic moduli in an earlier elastic solution Some of the most popular methods for measuring

(331 were replaced with complex moduli; experi- material damping have been reviewed and their l imita-mental and theoretical results were in agreement, tions described [38]. Some currently used dampingThe correspondence principle has been extended to test techniques have also been discussed. Dynamicanisotropic fiber-reinforced composites [341; the stiffness of a material is generally found by substi-elastic moduli from previous work [201 were re- tuting measured data on specimen geometry, mass

t placed by complex moduli. The key simplifying as- density, resonant frequency, and mode number insumptions were that the fibers are elastic and that the eigenvalue equation for the specimen and solvingthe matrix is elastic in dilatation but viscoelastic in for the effective elastic modulus This straightforwardshear. An expression for the complex flexural modu- procedure presents few difficulties. Measurement

lus of a cross-ply laminated beam has been obtained of damping of the material, however, is more dif-[351. The Hashin-Rosen random array model was ficult. The principal problem involves reducing theused for the plies, and it was assumed that plane extraneous energy losses in the apparatus to ansections remain plane during flexure. it has been acceptable level. The experimentalist must considershown (361 that the effective moduli for visco- support friction, air drag, acoustic radiation, andelastic composites can be related to phase properties transducer mounting if the Jamping to be measuredand that all results from elasticity theory, including in the system is primarily material damping [35].bounds, can be applied to viscoelastic materials with It is useful to calibrate the system with a materiala correspondence principle. (It is appropriate to note having known damping [35, 39, 40]. Aluminum is anagain that the elastost3tic theories of micromechanics excellent calibration material because its damping atand their viscoelastic counterparts may be used to low stress levels can be accurately predicted withdetermine dynamic properties only if the vibrational the Zener thermoelastic theory [39]

1]

Most damping measurements rely either on decay, Schultz and Tsai [44] used decay and band width

resonant dwell, or band width. The decay test mea- techniques to test aluminum and glass fiber-rein-sures free vibration decay of a specimen after its forced plastics in flexurai vibration. The damping inrelease from an initial displacement [41, 421. The unidirectional beams increased with increasingresonant dwell test and the band width test are forced frequency; dynamic elastic moduli did not correlatevibration techniques. The resonant dwell test involves well with static elastic moduli. When the techniquemeasurements of the resonant mode shape of the was extended to cross-ply laminates, however, damp-specimen [35, 39, 40, 43]. In the band width test ing decreased, leveled off, and increased with increas-the predicted shape of the amplitude frequency ing frequency [451. The tests were conducted in air,function is determined; the resonant peak and the so that some of the inconsistency could have resultedamplitudes at frequencies below and above resonance, from air damping. The elastic moduli of laminatesusually at the two half-power points [43, 44], are predicted from measured ply properties agreedmeasured. A fourth test that has been used success- fairly well with measured values of the composite;fully on viscoelastic materials is the impedance predicted laminate damping values were significantlytechnique [53, 54]. Techniques in which the speci- less than messured values.men is excited are generally applicable to a varietyof configurations. Beam specimens excited in flexure An apparatus for measuring the complex flexuralare most commonly used, however, because flexure modulus of aluminum and carbon fiber compositesis probably the most important mode of structural at different temperatures and amplitudes has beenvibration. Most structures have flexural members developed [46, 47]. A resonant dwell technique waswhose resonant frequencies generally lie within the used to test a free-free beam in vacuo. Considerablefrequency range of common sources of excitation. effort was devoted to the reduction of extraneousBar specimens excited in torsional or extensional energy losses in the apparatus. Data on damping atmodes have also been used various stress levels were presented, but no conclu-

sions were reported. This apparatus was also used toExperimental Results study the effects of fiber orientation and laminateMuch of the experimental work on the dynamic geometry [48).properties of composite materials has been concernedwith measuring frequency and temperature depen- In tests of graphite reinforced epoxy beams throughdence. Typically, a long, slender beam specimen the eighth mode, it was found that resonant frequen-is tested at successively higher frequencies either by cies of the higher modes were markedly lower thanexciting higher modes or by reducing the length of those predicted if shear deformation is neglectedthe beam. Both aluminum and fiber-reinforced plastic [49]. Timoshenko's beam theory accounted reason-beams have been tested by decay and resonant dwell ably well for the dependence of experimental reson-techniques [40]. Tests were conducted in vacuo to ant frequencies on mode number, slenderness ratio,minimize air damping; the results with aluminum and modulus ratio. The effects of resin type, degreewere in good agreement with the Zener theory. of cure, and fiber type on damping in glass fiber-The damping measured in fiberglass and boron- reinforced plastic beams at different temperaturesreinforced plastics was greater in the resonant dwell and frequencies have been investigated [50]. Wrighttests than in the decay tests; results with elastic [51] conducted decay tests on epoxy beams rein-moduli showed the reverse trend. This led to some forced with one type of glass and four types of car-doubts about the technique by which the resonant bon fibers Work has been done on dynamic proper-dwell specimens were mounted. Air damping was ties of fiber-reinf(,rced metals [52]; decay testsshown to be significant at low frequencies and were conducted on tungsten fiber-reinforced coppercorresponding large amplitudes. Paxson [43], in a specimens in flexure It was found that damping wascontinuation of earlier work [40], tested unidirec- independent of strain amplitude but was related totional and cross-ply laminated boron-reinforced the residual strains induced by rolling.plastics with resonant dwell, band width, and decaytechniques in vacuo. Agreement among the threemethods was poor.

12

A flexural resonant dwell technique has been used to tests of cross-ply laminates showed that, as the ulti-test unidirectional and cross-ply fiberglass-reinforced mate fracture strain of the transverse plies (thoseepoxy beams in vacuo [35]. The effects of configur- plies perpendicular to the loading direction) wasation and vibration frequency were studied both exceeded, the slope of the stress-strain curve wasexperimentally and analytically for small amplitudes, significantly reduced [62]. The reduction in stiff-large amplitude effects were studied experimentally. ness was accompanied by matrix cracking along theExperimentally-determined complex moduli at small direction of the transverse fibers. Gibson [35]amplitudes were in good agreement with those has reported that the damping in cross-ply laminatespredicted by the Hashin-Rosen random array model in flexural vibration is independent of amplitude if

[20, 34] and were within the bounds predicted by the maximum strain amplitude does not exceed thethe Hashin variational principle [18]. fracture strain of the transverse plies. After the

fracture strain has been exceeded, however, dampingThe complex moduli from beam vibration tests have increases and remains near the higher level even afterbeen used to predict the velocities and attenuation unloading and reloading. Corresponding reductionscoefficients of elastic waves traveling in fiber coin- in stiffness do occur, but damping is far more sensi-posites [55, 56]. Wave propagation was along the tive to microstructural damage than is stiffness.fiber axes of unidirectional material. The dynamic Cracking- of the fiber-matrix bond in transverseresponse of fiber-reinforced viscoelastic materials plies is apparently controlled by strain concentra-subjected to pulse loading along the fiber direction tion in the matrix.has been measured [57]; the measured responsewas compared with that predicted by effective modu- The attenuation of ultrasonic waves in unidirectionallus theories [20, 34] and by continuum theories and cross-ply specimens under static uniaxial stress[7, 101. The accuracy of the models depended upon has been studied [63]. It was found that attenuationinternal geometry and fiber packing geometry. increases rapidly above a certain threshold stress and

increases more rapidly for cross-ply laminates thanModes of vibration other than flexural vibration have for unidirectional material. It was noted that perman-been investigated. Carbon and glass fiber-reinforced ent damage apparently occurs because the attenua-plastics have been tested in torsion and in flexure tion does not return to its initial value after unload.[47]. Lifshitz [58] used decay and pulse attenuation ing. A sharp increase in acoustic emissions at the kneetechniques to study the dynamic torsional and of the stress-strain curve has been reported for boron-extensional moduli of pure epoxy and of unidirec- epoxy-reinforced aluminum specimens [64]. Heretional fiber composites. He showed that the proper- again, the loading was static, but some failure mech-ties of the epoxy matrix affect torsional moduli anism was associated with the reduction in stiffness.more than extensional moduli and that extensionalmoduli are essentially independent of frequency and The shear modulus and damping of carbon fiber-temperature. A torsional pendulum apparatus has reinforced epoxy rods in torsional vibration has beenbeen used to study the damping of boron-epoxy measured before and after cracks were induced bycomposites at elevated temperatures [59]. Damping static twisting of the rods to shear failure [651.in glass-epoxy and boron-aluminum composites has It was found that damping increased and the shearbeen measured by observing the decay of exthsional modulus decreased after cracks were induced. Damp-vibrations [60]. ing was much more sensitive to crack damage than

was the shear modulus. Schultz and Warwick [661If vibration amplitudes in fiber-reinforced composites used equipment similar to that used by Schultz andare sufficiently large, nonlinear inelastic behavior Tsai [44, 45] to measure the complex moduli ofmay be caused by microstructural damage [36, 611. glass fiber-reinforced epoxy beams as a function ofBecause this damage is usually permanent, dynamic fatiguing time They found that the amount ofstiffness and damping oroperties will exhibit perma- fatigue cracking correlated reasonably well withnent changes. The behavior of cross-ply laminates at the amount of reduction in stiffness and the amountlarge strains is of particular interest Static tensile of increase in damping. Amplitude effects were not

studied.

13

According to Broutman and Krock [1] transverse REFERENCESplies crack as a result of strain concentration in the

matrix due to the fiber packing arrangement. A sim- 1. Broutman, L.J and Krock, R.H., Modern

pie analysis for estimating the strain concentration Composite Materials, Addison-Wesley (1967).

for a regular fiber array has been developed (671.Matrix cracking is only one of several mechanisms 2. Broutman, L.J. and Krock, R.H., Compositeby which composite materials can fail. According to Materials, 1 -8, Academic Press (1975).Salkind [68), composites may fail by matrix crack-ing, delamination, fiber failure, interface debonding, 3. Jones, R.M., Mechanics of Composite Mater-and other mechanisms, either separately or in com- ials, McGraw-Hill (1975).bination. Salkind has proposed that, because the lossof stiffness in composite materials can result in struc- 4. Garg, S.K., Svalbonas, V., and Gurtman, G.A.,tural failure long before complete fracture occurs, Analysis of Structural Composite Materials,the fatigue failure criterion should be the number of Marcel Dekker Co. (1973).cycles to a given change in stiffness, rather than thenumber of cycles to fractures. As has been shown by 5. Bert, C.W. and Francis, P.H., "Compositeseveral of the references, the internal damping in Material Mechanics: Structural Mechanics,"fiber-reinforced composite materials is far more sensi- AIAA J., 12 (9), pp 1173-1186 (Sept 1974).tive to these failure mechanisms than is stiffnessso that damping measurements could be valuable 6. Tsai, S.W., Mechanics of Composite Materials,tools for the early detection of structural damage. AFML-TR-66-149, Parts I and It (1966).

7. Herrmann, G. and Achenbach, J.D., "WaveCONCLUSION Propagation in Laminated and Fiber-Reinforced

I . Composites," in Mechanics of Composite

We have attempted to present a balanced picture of Materials, Proc. 5th Symp. Naval Struct.

recent experimental and analytical efforts to charac- Merh, Pron 5th (170.

terize the dynamic mechanical properties of fiber- Mech., Pergamon Press (1970).

reinforced composite materials. This is a rapidly 8. Achenbach, J.D. and Herrmann, G., "Disper-growing field. most of the work cited has been done sion of Free Harmonic Waves in Fiber-Rein-within the past decade. Although much experimental forced Composites," AIAA J., 6 (10), pp 1832-and analytical work has been done, more comparisons 1836 (Oct 1968).of measured and predicted properties are needed toevaluate the various micromechanical theories. For 9. Sun, C.T., Achenbach, J.D., and Herrmann, G.,example, more definitive criteria could be estab- "Continuum Theory for a Laminated Medium,"lished on the limits of validity of effective modulus J. Appl. Mech., Trans. ASME, 35 (3), Ser. E,theories for analyzing the dynamic behavior of pp 467-475 (Sept 1968).composite structures. The actual mechanisms ofdamping in composite structures should be studied 10. Bedford, A. and Stern, M., "On Wave Propa-more extensively, both experimentally and analy- gation in Fiber-Reinforced Viscoelastic Mater-tically. Because damping is so sensitive to microstruc- ials," J. Appl. Mech., Trans. ASME, 37, pptural damage, damping measurements could provide 1190-1192 (1970).valuable quality control and reliability information

11. Ross, C.A and Sierakowski, R.L., "ElasticWaves in Fiber-Reinforced Composites," Shock

ACKNOWLEDGMENI' Vib Dig., 7 (1), pp 96-107 (Jan 1975).

Financial assistance for this work has been provided

at different times by the U.S. Air Force Materials 12. Moon, F.C., "A Critical Survey of Wave Propa-

Laboratory, the University of Minnesota Graduate gation and Impact in Composite Materials,"

School, and the Engineering Research Institute at NASA CR-121226 (1973).

Iowa State University

14

13. Achenbach, J.D., "Waves and Vibrations in 24. Hahn, H.T. and Tsai, S.W., "Nonlinear ElasticDirectionally Reinforced Composites," in Corn- Behavior of Unidirectional Composite Lamin-posite Materials, (Broutman and Krock, Eds.) ae," J. Composite Matl., 6, pp 102-118 (Jan2, Academic Press (1975). 1973).

14. Chamis, C C and Sendeckyj, G.P., "Critique 25. Hahn, H.T , "Nonlinear Behavior of Laminatedon Theori6s Preoicting Thermoelastic Proper- Composites," J. Composite MatI., 6, pp 257,ties of Fibrous Composites," J. Composite 271 (Apr 1973).Mat[., 2, pp 332-358 (July 1968).

26. Rehfield, L.W., "Nonlinear Considerations in15. Hashin, Z., "Theory of Mechanical Behavior the Vibration of Resin Matrix Composite

of Heterogeneous Media," Applied Mechanics Structures," Proc. 17th AIAA/ASME/SAESurveys, pp 263-275, Spartan Books (1966). Struc., Struc. Dyn., Mati. Conf., King of

Prussia, PA, pp 168-173 (May 1976).16. Paul, B., "Prediction of Elastic Constants of

Multiphase Materials," Trans. Metallurgical Soc. 27. Lazan, B.J., Damping of Materials and MembersAIME, 218, pp 36-41 (Feb 1960). in Structural Mechanics, Pergamon Press

(1968).

17. Hashin, Z. and Shtrikman, S., "On SomeVariational Principles in Anisotropic and Non- 28. Laian, D.J,, "Damping Properties of Materials,homogeneous Elasticity," J Mech. Phys. Members, and Composites," Applied MechanicsSolids, 10, pp 335-342 (1962). Surveys, pp 703.715, Spartan Books (1966).

18. Hashin, Z., "On Elastic Behavior of Fibre 29. Bert, C.W., "Material Damping: An Introduc-Reinforced Materials of Arbitrary Transverse tory Review of Mathematical Models, MeasuresPhase Geometry," J. Mech. Phys Solids, 13, and Experimental Techniques," J. Sound Vib.,pp 119-134 (1965). 29 (2), pp 129-153 (July 1973).

19 Hill, R., "Theory of Mechanical Properties of 30. Snowdon, JC., Vibration and Shock in Damp-Fibre-Strengthened Materials. I. Elastic Behav- ed Mechanical Systems, John Wiley & Sonsior," J Mech Phys. Solids, 12, pp 199-212 (1968).(1964).

31. Lee, E.H., "Stress Analysis in Viscoelastic20. Hashin, Z. and Rosen, B.W., "The Elastic Mo- Bodies," Quart. AppI. Math., 13 (2), pp 183-

duli of Fiber-Reinforced Materials," J Appl 190 (July 1955).Mech., Trans ASME, pp 223-232 (June 1964)Errata, p 219 (March 1965). 32. Hashin, Z., "Complex Moduli of Viscoelastic

Composites I. General Theory and Applica-21 Adams, D.F. and Tsai, S.W., "The Influence of tion to Particulate Composites," Intl. J. Solids

Random Filament Packing on the Transverse Struc ,6, pp 539-552 (1970)Stiffness of Unidirectional Composites," J.Composite Matl ,3, p 368 (July 1969). 33. Hashin, Z., "The Elastic Moduli of Hetero-

geneous Materials," J. Appl. Mech., Trans.22 Adams, D F. and Doner, D.R., "Transverse ASME, 29, p 143 (1962).

Normal Loading of a Unidirectional Compcs-ite," J Composite Matl., 1, pp 152-164 (Apr 34. Hashin, Z., "Complex Moduli of Viscoelastic1967). Composites II Fiber Reinforced Materials,"

Intl J Solids Struc , 6, pp 797-807 (1970).23. Halpin, J C and Tsai, S W., "Effects of Envir-

onmnental Factors on Composite Materials,"AFML-TR67-423 (June 1969)

15

35. Gibson, R.F., "Elastic and Dissipative Pro- 45. Schultz, A.B. and Tsai, S.W., "Measurementsperties of Fiber-Reinforced Composite Mater- of Complex Dynamic Moduli for Laminatedials in Flexural Vibration," Ph.D. Dissertation, Fiber-Reinforced Composites," J. CompositeUniv. Minn., Minneapolis (Dec 1975). MatI., 3, pp 434-443 (July 1969).

36. Schapery, R.A., "Viscoeiastic Behavior and 46. Adams, R.D. and Bacon, D.G.C., "MeasurementAnalysis of Composite Materials," Texas of the Flexural Damping Capacity and Dy-A&M Univ. Mech. Mati. Res. Ctr. Rep. No. namic Young's Modulus of Metals and Rein-MM 72-3 (Aug 1972). forced Plastics," J. Phys. D (AppI. Phys.),

6, pp 27-41 (1973).37. Chang, S. and Bert, C.W., "Analysis of Damping

for Filamentary Composite Materials," Com- 47. Adams, R.D. and Bacon, D.G.C., "The Dynam-posite Mati. Engr. Des., ASM, pp 51-62 (1973). ic Properties of Unidirectional Fibre Reinforced

Composites in Flexure and Torsion," J. Com-38. Plunkett, R., "Measurement of Damping," posite Mati., 7, pp 53-66 (Jan 1973).

in Structural Dampin-g (Rusicka, ed.), ASME,pp 117-131 (Dec 1959). 48. Adams, R.D. and Bacon, D.G.C., "Effect of

Fibre Orientation and Laminate Geometry39. Granick, N. and Stern, J.E., "Material Damping on the Dynamic Properties of CFRP," J.

of Aluminum by a Resonant Dwell Technique," Composite Matl., 7, pp 402-428 (Oct 1973).NASA TN D-2893 (1965).

49. Dudek, T.J., "Young's and Shear Moduli of40. Mazza, L.T., Paxson, E.B., and Rodgers, R.L., Unidirectional Composites by a Resonant

"Measurement of Damping Coefficients and Beam Method," J. Composite Mail., 4, pp 232-Dynamic Modulus of Fiber Composites," 241 (Apr 1970).USAAVLABS Technical Note 2, U.S. ArmyAviation Material Laboratories, Fort Eustis, 50. Ehrenstein, G.S. and F6rstor, R., "DampingVA (Feb 1970). Properties of Glass Fibei -Reinforced Plastics,"

Transl. Proc. 7th Ann. Mtg. Working Group41. Baker, W.E., Woolam, W.E., and Young, D., Reinforced Plastics, Freudenstadt, W. Ger.,

"Air and Internal Damping of Thin Cantilever pp 33-1 to 33-7 (Oct 1968).Beams," Intl J. Mech. Sci., 9, pp 743-766(1967). 51. Wright, G.C., "The Dynamic Properties of Glass

and Carbon Fibre Reinforced Plastic Beams,"42. Hagel, W C and Clark, J.W., "The Specific J. Sound Vib., 21 (2), p 205 (1972).

Damping Energy of Fixed-Fixed Beam Speci-mens," J Appl. Mech., Trans. ASME, pp 426- 52. Yarnane, T., Umakoshi, Y., Tomishima, M.,430 (Sept 1957). and Kazuyoshi, N., "A Study of Internal

Friction in Fiber-Reinforced Copper Com-43. Paxson, E.B., Jr , "Real and Imaginary Parts posites (Tungsten Wire)," in Mech. Behavior

of the Complex Viscoelastic Modulus for Matl., Soc Mati. Sci., V, p 220, Kyoto, JapanBoron Fiber Reinforced Plastics," J. Acoust. (1972).Soc Amer., 57 (4), pp 891-898 (Apr 1975).

53. Edwards, J.L. and Hicks, D.R., "A Mechanical44. Schultz, A.G. and Tsai, S W., "Dynamic Moduli Impedance Technique for the Measurement of

and Damping Ratios in Fiber.Reinforced the Dynamic Properties of Materials," Proc.Composites,' J Composite Matil , 2 (3), pp 1971 AFSC Sci. Engr. Symp., II, Dayton, OH368-379 (July 1968) (Oct 1971).

16

54. Laird, G.W. and Kingsbury, H.B., "A Method 62. Lavengood, R.E. and Ishai, 0., "The Mechani-of Determining Complex Moduli of Viscoelas- cal Performance of Cross-Plied Composites,"tic Materials," Exptl. Mech., 13 (3), pp 126- Polymer Engr. Sci., 11 (3), pp 226-232 (May131 (Mar 1973). 1971).

55. Tauchert, T.R. and Moon, F.C., "Propagation 63. Tauchert, T.R. and Hsu, N.N., "Influence ofof Stress Waves in Fiber-Reinforced Com- Stress upon Internal Damping in a Fiber Rein-posite Rods," Proc. 11 th AIAA/ASME Struc., forced Composite Material," J. CompositeStruc. Dyn., Mati. Conf., Denver, CO, p 176 Matl.,7, pp 516-520 (Oct 1973).(Apr 1970).

64. Henneke, E.G., Herakovich, C.T., Jones, G.L.,56. Jones, E.R., Sierakowski, R.L., and Ross, C.A., and Renieri, M.P., "Acoustic Emission from

"Damping Measurements of a Controlled Composite Reinforced Metals," Exptl. Mech.,Composite Material,' J. Acoust, Soc. Amer., 15, p 10 (Jan 1975).57 (6), Part II, pp 1465-1472 (June 1975).

65. Adams, R.D., Flitcroft, J.E., Hancox, N.L.,57. Sutherland, H.J. and Calvit, H.H., "A Dynamic and Reynolds. W.N., "Effects of Shear Damage

Investigation of Fiber-Reinforced Viscoelastic on the Torsional Behavior of Carbon FibreMaterials,' Exptl. Mech., 14 (8), pp 304-310 Reinforced Plastics," J. Composite Matl.,(1974). 7' pp 68-75 (Jan 1973).

58. Lifshitz, J.M., "Specimen Preparation and 66. Schultz, A.B. and Warwick, D.N., "VibrationPreliminary Results in the Study of Mechanical Response: A Nondestructive Test for FatigueProperties of Fiber Reinforced Material," Crack Damage in Filament-Reinforced Com-MML Rep. No. 15, Technion, Israel Inst. posites," J. Composite Mati., 5, pp 394-404Tech., Haifa (Jan 1969). (July 1971).

59. Schrager, M. and Carey, J., "Viscoelastic 67. Kies, J.A., "Maximum Strains in the Resin ofBehavior of Boron Fiber-Epoxy Resin Com- Fiberglass Composites," Naval Res. Lab.posites," Polymer Engr. Sci., 1.0, p 369 (1970). Rep. No. 5752 (1962).

60. Pottinger, M.G., "Materials Damping of Glass 68. Salkind, M.J., "Fatigue of Composites," inFiber Epoxy and Boron Fiber Aluminum Composite Materials- Testing and Design,Composites," Aerospace Res. Labs, Wright- ASTM-STP-497, pp 143-169 (1972).Patterson AFB, OH, NTIS-CSCL (1970).

61. Schapery, R.A., "Deformation and FailureAnalysis of Viscoelastic Composite Materials,"in Inelastic Behavior of Composite Materials,ASME-AMD, 12, pp 127-155 (Nov 1975).

17

FINITE ELEMENT MODELING OF NATURAL VIBRATION PROBLEMS

A. V. Krishna Murty*

Abstract. This is a review of finite element modeling matrix methods, and the matrix displacement meth-techniques including lumped parameter, transfer od. There are variations of this concept which do notmatrx, finite element displacement, finite element fit into this philosophy. These are not included in thisforce, hybrid end quadratic elgenvalue method Ap. review. For instance the finite difference method mayplications of the methods to natural vibration pro. be ccnsidered an exception to the finite elementblema are given, method. In the finite difference method, the equili-

brium equation at a grid point involves a set of valuesAny structure will undergo distortion and oscillations at a specified set of neighboring grid points. Overlap-if the proper force is applied. If the amplitude is ping submains are used to form the governing equa-sufficient, high stresses and damage to the structure tion at a point. The emphasis in this method is oncan occur. The need to avoid structural damage forming equations of equilibrium directly at a point;resulting from vibration was the motivition for the thus, the finite difference method does not involve anstudy of structural vibrations. Natural frequencies assembly of subdomains. Instead, equilibrium and/orand mode shapes have been used in the determin- compatibility is satisfied in an average sense at a selec-ation of the dynamic response of structures. The ted set of points.desire to achieve as simple an analytic procedureas possible and accurate results has motivated the Most finite element methods currently available candevelopment of many analytic models. The objective be placed under one of the following: lumped para-is to model the problem as close to reality as possible. meter methods, transfer matrix methods, finiteIn practice, for such structures as plates and shells, element displacement methods, finite element forcecontinuum treatment has been used. Such structures methods, hybrid methods, and quadratic eigenvalueas frames and trusses can be handled with simpler methods.discrete models. Discrete models often can be usedto represent complex structural configurations. Lumped Parameter MethodsAmong discrete models reported in the literature are The lumped mass method [121 - 123, 1321 is perhapsthose involving lumped mass, lumped inertia forces, the simplest approach to the analysis of naturalfinite differences, transverse matrix, and the finite vibrations. The mass of the structure is lumped atelement. This paper is intended as a review of the a set of selected grid points Half the mass of theliterature of modeling with finite elements. elements that meet at a grid point may be used as

the criterion for lumping the masses. The stiffnessIn the formulation of a natural vibration problem influence coefficients with respect to the sameusing finite elements the structure is treated as an grid points are obtained throogh either a ,,eparateassembly of subdomains called elements. A typical analysis or an experiment 1he governing equationfinite element idealization is shown in the Figure. is of the type shown in Equ. 1i,,n (1).Each element is based on a physical or mathematicalprinciple that allows calculation of quantities reore- K] It'- - [M] 0senting various properties of the struIcture nr(.e (1)displacement relationships, elastic and .iertia forces,and strain and kinetic energies The governing equa- The accuracy of the results depends upon the i.;ethodtions for the complete structure are derived using of lumping the masses used. For example, the inertiaan appropriate assembly procedure Boundary condi- force in each element may be lumped at the gridtions are defined,, and natural frequencies and mode points. However,, the representation of strain energyshapes aie obtained Matr-ces are used in some and kinetic energy is somewhat inconsistent. The

-methods, including the lumped mass method, transfer representation is akin to using one displacement

Ass-ciate Professor, Dept. of Aeronautical Engineering, Indian Institute of Science, Bangalore, India

19

FIG. 1. FINITE ELEMENT IDEALIZATION OFA TRIANGULAR PLATE.

20

field in the element for the derivation of the stiffness andmatrix (strain energy) and a different displacement Nfield in the same element for the derivation of the N= T [Ti] {dt = I fda (5)mass matrix (kinetic energy). !t is because of this 11,2,..inconsistency that the nature of the bound of the [T] is a transcendental matrix containing the eigen.eigenvalues cannot be predictrd. The rate of conver- value w . A value of w that satisfies prescribedgence and the nature of the bound depend upon the values in 1 and W N is chosen by trial andmethod of lumping. It has been found [581 that, error or a graphical procedure. The success of thisin the case of torsional oncillations of a shaft, the method rests largely on the development of theeigenvalue is estimated as a lower bound with the transfer matrix ITI. This method is particularlyrate of convergence as 1/m, m being the number of suited to dimensional problems. The matrices areelements used. The element mass matrix [1211 relatively small in comparison with those of the

r finite element displacement method. The finite2 1 element displacement method has replaced transfer

[in, --- (2) matrix methods in many applications -- especiallyin natural vibration problems.

gives upper bounds to eigenvalues, but the rate ofconvergence is the same as 1/m2 . Another element Finite Element Displacement Methodsmass matrix A finite element displacement model* was introduced

into vibration analysi4 in the 1960s by Archer (151

!ma " and Argyris [17]. Because this model has become12 1" 5 (3) the most versatile tool for analyzing natural vibration

problems, it is described in some detail. The methodproduces a lower bound to the eigenvalue with the is derived from the Rayleigh-Ritz method, whichrate of convergence being 1/in. Similar trends have may be stated asalso been noted in the finite element analysis ofplates (135]. The inconsistencies in the lumped 6 (U-T) 0parameter methods can thus become an advantage (6)by the appropriate choice of mass matrices. In theliterature the fact that the mass matrix governs the U is the strain energy, and T is the kinetic energy.rate of convergence and the nature of bounds has The structure is divided into a network of connectedbeen brought out by numerical correlative studies domains, or elements. In each element, the displace-and by the use of simple classical examples for ment field I is chosen in terms of the elementwhich error estimates are possible. nodal displacement vector 161 as

Transfer Matrix Methods IjI = (NJ f6Transfer matrix methods (37, 50, 51, 93, 144, 150] (7)have been used in vibration analysis since 1921 whenHolzer developed his tabular method for torsional [N] is the matrix of shape functions. The vector ofvibration problems. Later Mykelstad [247], and strains lei in the element may be written asProhl [134], adapted the Holzer Method to beamproblems. This method will be described for a beam lei (LI l+1, [B] 161problem. The beam is divided into several elements. (8)At each nodal point a state vector j is defined;it consists of the displacements and forces at that [LI is an appropriate diffeiential operation matrix.point A transfer matrix [TI relating the state Equations (7) and (8) are used to determine thevector at two neighboring points is developed using strain energy U(e) and kinetic energy T(e) in thethe governing differential equation. Thus element

U (e) = -181 T [k] 161

1 11= [T] , 2 (9)(4)

*Most of the bibliography deals with this method

21

Finite Element Force Methods

T(e) 6 IT [ml 16 1 Finite element force methods [60, 215 - 2191 are2 (10) based on complementary energy principles. Most of

the work in this area has been done by Tabarrok[k] and [m) are the element stiffness and mass 1216 - 220] and his associates. A variational principlematrices defined as that is complementary to Hamilton's principle has

been used to calculate plate frequ-encies. The method[k] = f [B]T [D] [B] dV (11) utilized the variational principle in the form

v

[m] =f [NIT I [NJ dV (12) 6 ? (V-T) dt=O (18)v to

(D] is the elasticity matrix, and p is the mass T is kinetic energy expressed in terms of the d'Alem-density of the material. The total strain energy U and berts impulses e asthe kinetic energy T are obtained by adding the 1,contributions of each element T = -- f e1 dx dy (19)

1 itIT fKI It= T [K] (13) V is the complementary strain energy. This formu-

.IIT [MIlation shows that the plate has an infinite number ofT=2 € T M] ' (14) zero frequency modes. It is essential to eliminat, all

of these modes in order to predict the natire ofwhere bound of the frequency. It has been demonstrated

that the complementary energy formulation might[K,M] = [alT -,,mi [a] (15) result in natural frequencies lower than the true

natural frequencies of the plates. In 1973, Tabarrok[a] is the displacement transformation matrix and and Sodhi [220] generalized the stress functionrelates the elemental nodal displacements 6 61 and procedure and used Toupin's variational principle.the global or structural displacements as Four conforming finite element models have now

been developed and applied to linear bending vibra-61 = [a] It' (16) tion problems of plates. The convergence to the

eigenvalues may proceed either from below or fromEquations (13) and (14) are used in equation (6) above, depending upon the order of approximationsto obtain the governing equation employed for the stress functions. It has been bug-

gested that the procedure could be extended to any(K] l}- O2 [M] I } =0 (17) problem for which stress functions have already

been developed. It has also been noted that the stressfrom which eigenvalues and eigenvectors are deter- function formulation is not always advantageous, es-mined. pecially with regard to suppressing any frequency

modes, a concept introduced by Tabarrok andWith finite element displacement models the dis- Sedhiplacements are treated as unknowns. Variations ofthe method described above are possible. Attempts Hybrid Methodshave been made [185] to construct finite element The combination of two or more methods to mini-models based on Temple's successive approximation mize disadvantages and maximize advantages isscheme Complete correspondence between finite called a mixed, or hybrid, method [36, 59, 96, 97,element models and a modified Rayleigh-Ritz method 126, 149, 175, 206]. Finite element methods thathas been established [177]. This method has proved compromise certain basic aspects of the parentuseful for one-dimensional problems continuum method are hybrid methods. Some hybrid

methods concentrate on a specific advantage. Thepossibility that the force method might yield lowerbounds to eigenvalues and the basic difficulties in

22

formulating the vibration problem in terms of forces error. When two real values are obtained, they boundsuggests that hybrid methods that combine energy the natural frequency. In either case, therefore, anand complementary energy methods might be useful. approximation of the eigenvalue and an indication ofRao et al.,[77, 78;, attempted to formulate vibration error are simultaneously provided. Quadratic eigen-problems using complementary energy. They treated value methods are probably of greatest value in study-the problem in terms of forces. Shape functions have ing basic problems. Their application to complex pro-been used to satisfy the equilibrium equations (36]. blems has not been established. In particular, stan-The shape functioiis contain the eigenvalue, an itera- dard eigenvalue routines cannot be used to obtaintive method of solution is thus necessary, With an- numerical solutions; thus, special programming isother method [591, the stiffness matrix is derived by necessary.assuming suitable forms for stresses and boundarydisplacements, The mass matrix is based on an inde- Other Methodspendently chosen displacement field in the element. Any continuum method can be used to develop aHybrid elements have been used to study cylindrical finite element method. Most methods currently inshell vibrations [97, 981. In another case [1491, use depend on a minimization principle. When thismasses have been represented by modified potential is not possible or convenient, it is useful to developenergy. The results have been more accurate than models based on methods for the solution of differ-those obtained with conventional methods. The ential equations such as the Galerkin method 1126,feasibility of a Pian-type hybrid model has been 175, 176], collocation (175, 1761, or an ortho.outlined [126]. These methods have not yet been gonality method. The differential equation governingfully explored; considerable further study is needed the free vibration problem may be written asbefore viable hybrid models will be available forvibration problems. L(/)- V L2(A) = 0 (22)

Quadratic Eigenvalue Methods L, and L2 are differential operators, and w is theTwo types of formulation leading to a quadratic eigenvalue. In a one-dimensional problem, the dis.eigenvalue problem of the type placement distribution in the i-th element may be

taken as

(A] I} + O2 ! } + [C] { = (20) [i [N] 16 (23)

have been reported (126, 179). Prezemieniecki'swork (179] is based on the premise that shape [N] is the matrix of shape functions and ele-functions must contain the frequency. The problem ment nodal displacements. W1 is substituted in theis of the type governing differential equation to obtain the expres-

sion for the error in the differential equation in the[Al] - o' [A2 ] - W4 [A3] .... J c4'0 (21) i-th element; eibecomes

Natural frequencies have been worked out retaining ei = (LI N] - ( 6 (24)terms up to [A3] Results have confitmed that theaccuracy of the natural frequencies and mode shapes With the Galerkin procedure the integral of the pro-might be improved by retaining the term with the duct error in the differential equation and the com-frequency to the fourth power. This formulation ponents of the displacement functions are set equalcontains negative eigenvalues, the reasons for their to /ero.occurrence and their influence have not yet beenestablished Another quadratic eigenvalue problem (Ek] w2'I- (Ern =0 (25)is based on the least square method (126] . The re-silts of this procedure are either a complex pair of whereeigenvalues or two real values corresponding to each ( ] lnatural frequency. When complex numbqrs are [8 e] a (26)obtained, the modulus is an approximation of the r =frequency, and the imaginary value is an indication of rEk] (a] ekj (a] (27)

23

and 4. Ahmed, K.M., "Dynamic Analysis of Sandwich

[em) [a] T em [a) Beams,'J. Sound Vib., 21 (3), pp 263-276i (28) (1972).

with

[eki] = fNT [L NJ dt b. Ahmad, S., Anderson, R.G., and Zienkiewicz,e (29) O.C., "Vibration of Thick Curved Shells with

and Particular Reference to Turbine Blades,"(emi] = [NJT [Li NJ d J. Strain Anal., 5, pp 200-206 (1970).e 130)-

The eigenvalues and eigenvectors can be computed 6. Jennings, Alan, "Mass Condensation andfrom equation (25). Small amplitude vibration Simultaneous Iteration for Vibration Pro-problems have been studied using this method [126, blems," Intl. J. Numer. Methods Engr., 6,

1761. Only linear eigenvalue problems have been pp 543.552 (1973).studied thus iar using these methods, but theyare likely to prove useful for studying nonlinear 7. Jennings, Alan and Orr, D.R.L., "Applicationoscillations, of the Simultaneous Iteration Method to Un-

damped Vibration Problems," Intl. J. Numer.Methods Engr.,3, pp 13.24 (1971).

CONCLUSIONS8. Celia, Aldo, Comments on "Rates of Change of

Some finite element models have been compared Eigenvalues and Eigenvectors," AIAA J., 7,in the Table. The Rayleigh.Ritz finite element me- p 1655 (1969).thod is the most convenient and efficient one, atleast, insofar as linear natural vibration problems are 9. Berman, Alex and Flannely, William G., "The-concerned. Other methods may be more efficient ory of Incomplete Models of Dynamic Struc-but have some disadvantages that must be eliminated. tures," AIAA J.,9, p 1481 (1971).

10. Allik, H. and Hughes, T.J.R., "Finite ElementAcknowledgment Method for Pie.oelectric Vibration," Intl. J.The author takes this opportunity to thank Mr. K. N. Numer. Methods Engr., 2, pp 151-159 (1970).Shivakumar, Mr T. S. R. Reddy, and Mr. S S. Mur-thy for their assistance in compiling the references. 11. Allik, H., Webman, K.M., and Hunt, J.T.,

"Vibrational Response of Sonar Transducer

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tion of Cylindrical Shells Partially Filled withLiquid," J. Sound Vib., 19 (1), pp 1-15 (1971). 140. Paz, Mario and Dung, Lam, "Power Series

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McGraw-Hill (1966). Engr., 9, pp 449-459 (1975)

130. Laursen, H L., Subinski, R.P., and Clough, R.S., 141. Martin, P.C., Introduction to Matrix Methods"Dynamic Matrix Analysis of Framed Struc- of Structural Analysis, McGraw-Hill (1966).tures," Proc. Fourth U.S. NatI. Conf. Mech.,1, pp 99-105 (1962). 142. Mason, V., "Rectangular Finite Element for

Analysis of Plate Vibrations," J. Sound Vib.,131. Leckie, F.A., "The Application of Transfer 7, pp 437-448 (1968).

Matrices to Plate Vibrations," Ing Arch.,32, p 100 (1963). 143. Matsumoto, K., "Vibration of an Elastica

Immersed in Fluid," Second U.S.-Japan Semin-132. Leckie, F A. and Lindberg, G M., "The Effect ar on Matrix Methods of Structural Analysis

of Lumped Parameters on Beam Frequencies," and Design, Univ. Alabama Press, pp 259-274Aeron. Quart., 14, pp 224-240 (1963). (1972)

133. Meirovitch, Leonard, "Comment on Non-zero 144. McDaniel,. T.J., "Dynamics of Non-circularFree-free Frequencies of Structures Idealized Stiffened Cylindrical Shells," J. Sound Vib.,by Matrix Methods," AIAA J., 8, p 1370 23(2), pp 217-227 (1972)(1970).

30

145. McMinn, S.J., Matrices for Structural Analysis, 156. Nickel, R.E. and Secor, G.A., "ConvergenceJohn Wiley and Sons (1962). of Consistently Derived Timoshenko Beam

Finite Elements," Intl. J. Numer. Methods146. Mei, C., "Coupled Vibrations of Thin-walled Engr., 5, pp 243-253 (1972).

Beams of Open Section Using the Finite Ele-ment Method," Intl. J. Mech. Sci., 12 (10), pp 157. Nosseir, T.A. and Dickinson, S.M., "The Free883-891 (1970). Vibration of a Model of a Car Body," J. Sound

Vib., 15, pp 257-268 (1971).147. Mei, C. and Yang, T Y., "Free Vibrations of

Finite Element Plates Subjected to Complex 158 Olson, M.D. and Lindberg, G.M., "Annular andMiddle Plane Force Systems," J. Sound Vib., Circular Finite Elements for Plate Bending,"23(2), pp 145-156 (1972). Intl. J. Mech. Sci., 12 (1), pp 17-33 (1970).

148. Mei, C., "Free Vibrations of Circular Mem- 159 Olson, M.D. and Lindberg, G.M., "Dynamicbranes under Arbitrary Tension by Finite Analysis of Shallow Shells with a DoublyElement Method," J. Acoust. Soc Amer., Curved Triangular Finite Element," J. Sound46, p 693 (1969) Vib., 19, pp 299-318 (1971).

149. Melosh, R.J. and Lang, T.E., "Modified Poten- 160. Olson, M.D. and Lindberg, G.M., "Vibrationtial Energy Mass Representations of Frequen- Analysis of Cantilever Curved Plates Using acy Prediction," P.FC.M.M.S.M.**, Wright- New Cylindrical Shell Finite Element," P.S.Patterson AFB, Dayton, OH, (1965). C.M.M.S.M.*, Wright-Patterson AFB, Dayton,

Oh, AFFDL-TR-68-150, pp 247-270 (1968).150. Mercer, C A. and Seavey, C., "Prediction of

Natural Frequencies and Normal Modes of 161. Paidoussis, M.P. and Lakis, A.A., "VibrationSkin Stringer Panel Rows," J. Sound Vib., Characteristics of Cylindrical Shells with6,pp 149-162 (1967). Several Axially Equi.spaced Constraints," J.

Sound Vib., 24, pp 51-62 (1972).,151. Monforton, G.,R. and Schmit Jr., L.A., "Finite

Element Analysis of Sandwich Plates and 162. Paul, P. Lynn and Balaurs, S. Dhillon, "Conver-Cylindrical Shells with Laminated Faces," gence of Eigenvalue Solutions in ConformingP.S.CM M.S M.,*, Wright-Patterson AFB, Day- Plate Bending Finite Elements," Intl. J. Numer.ton, OH, AFFDL-TR-68-150, pp 573-616 Methods Engr., 4, pp 217-234 (1972).(1968)

163. Pestel, E.C., "Dynamic Stiffness Matrix Formu-152 Murthy, V R. and Nigam, N.C , "Dynamic lation by Means of Hermition Polynomials,"

Characteristics of Stiffened Rings by a Trans- P.F.C.M.M.S.M.**, Wright-Patterson AFB, Day-fer Matrix Method,' J Sound Vib., 39, pp 237- ton, OH (1966).245 (1975).

153. Nagarala, V T , "Changes in Vibration Charac- 164. Pestel, E.C. and Leckis, F.A, Matrix Methodsteristics due to Change in Structure," J in Elastomechanics, McGraw-Hill (1963).Spacecraft Rockets, 7, pp 1499-1501 (1970).

165 Petyt, M., "Vibration of Curved Plates," J.154. Nagaraja, V.T , "The Influence of Concentrated Sound Vib., 15, pp 381-395 (1971)

Inertias on the Free Vibrations of Beams andShafts," Aeron Soc. India J., 23, p 145 (1971). 166. Petyt, M , "The Vibration Characteristics of

a Tensioned Plate Containing a Fatigue Crack,"155. Navaratna, D R , "Natural Vibrations of Deep J. Sound Vib., 8, pp 377-389 (1968).

Spherical Shells," AIAA J, 4, pp 2056-2058(1966)

31

167. Petyt, M. and Fleischer, C.C., "Free Vibration 179. Prezmieniecki, J.S., "Quadratic Matrix Equa-of a Curved Beam," J. Sound Vib., 18, pp 17- tions for Determining Vibration Modes and30 (1971). Frequencies of Continuous Elastic Systems,"

P.F.C.M.M.S.M. ", Wright-Patterson AFB, Day-168. Petyt, M. and Fleischer, C.C., "Vibration of ton, OH, pp 777-802 (1966).

32, pp 359-365 (1974). 180. Prezmieniecki, J.S., Theory of Matrix Struc-tural Analysis, McGraw-Hill (1968).

16.). Petyt, M. and Mirza, W.H., "Vibration ofColumn-supported Floor Slabs," J. Sound Vib., 181. Raju, I.S., Prakasarao, B., and Venkateswara21, pp 355.364 (1972). Rao, G., "Axisymmetric Vibrations of Linear-

ly Tapered Annular Plates," J. Sound Vib.,170. Petyt, M. and Mirza, W.H., "Dynamic Behav- 32, pp 507-512 (1974).

iour of In-line Shear Walls Connected byFloor Slabs," J. Sound Vib., 25, pp 349-357 182. Ramamurthy, V., "Application of the Simul-(1972). taneous Iteration Method to Torsional Vibra-

tion Problems," J. Sound Vib., 29, pp 331-340171. Petyt, M. and Mirza, W.H., "Vibration of (1973).

Asymmetrical Coupled Shear Walls," J. SoundVib., 27, pp 573-581 (1973). 183. Ramamurthy, V. and Ganesan, N., "Torsional

Vibrations of Prismatic Shells," J. Sound Vib,,172. Pipes, L.A., Matrix Methods for Engineering, 38pp 195-213(1975).

Prentice-Hall (1963).184. Ramsden, J.N. and Stoker, J.R., "Mass Conden-

173. Popplewell, N., "The Vibration of a Box-type sation: A Semi-automatic Method for ReducingStructure: I. Natural Frequencies and Normal the Size of Vibration Problems," Intl. J. Nu-Modes," J. Sound Vib., 14, pp 357-365 (1971). merMethods Engr., 1, pp 333-349 (1969).

174. Popplewell, N., "The Vibration of a Box-type 185. Rao, G.V., Rao, B.P., and Raju, IS., "Vibra-Structure: I1. Response to a Travelling Pres- tions of Inhomogeneous Thin Plates Using asure Wave," J. Sound Vib., 18, pp 521-531 High Precision Triangular Element," J. Sound(1971). Vib., 34(3), pp 444-445 (1974).

175. Prasad, K.S.R.K., "Finite Element Adaptations 186. Rao, A.K., Ramamurthy, T.S., Murty, A.V.K.,of Continuum Methods for Vibration Pro- and Rao, G.V., "Bounds and Error Controlblems,' Ph.D. Thesis, Indian Institute of of Eigenvalues," Theory and Practice in FiniteScience (1974). Element Structural Analysis, Tokyo Univ.

Press (1973).176 Prasad, K.S.R K. and Krishnamurty, A.V.,

"Galerkin Finite Element Method for Vibra- 187. Rao, S.S., "Finite Element Flutter Analysis oftion Problems," AIAA J, 11, p 544 (1973). Multi-web Wing Structures," J. Sound Vib.,

38, pp 233-244 (1975).177. Prasad, K.S R.K , Krishnamurty, A.V., and

Rao, A.K, "A Finite Element Analogue of the 188 Rao, S S., "A Study of Some Finite ElementModified Rayleigh-Rit Method for Vibration Ideali/ations for the Dynamic Analysis ofProblems," Intl. J Numer. Methods Engr, Aircraft Wing Structures," Aeron. Soc. India5, pp 163-169 (1972). J., 26 (3, 4) (1974).

178. Prentis, J M. and Leckie, F.A., MechanicalVibrations An Introduction to Matrix Meth-ods, Longmans, Green, and Co , Ltd , London(1963).

32

189. Rao, C.K., Gupta, B.VR., and Rao, D.L.N., 201. Craig, Jr., Roy R. and Bampton, Mervyn C.C.,"Torsional Vibrations of Thin-walled Beams "Coupling of Structures for Dynamic Analy-on Continuous Elastic Foundation Using the sis," AIAA J., 6, pp 1313-1319 (1968).Finite Element Method," Proc. First Intl.Conf. Finite Element Methods in Engineering, 202. Orris, Ruth M. and Petyt, M., "A Finite Ele-Coimbatore, India, pp 231-248 (1974). ment Study of the Vibration of Trapezoidal

Plates," J. Sound Vib., 27, pp 325-344 (1973).

190. Holzer, H., Torsional Vibration Calculations,

Springer Verlag, Berlin (1921). 203. Orris, Ruth M. and Petyt, M., "A Finite Ele-ment Study of the Harmonic Wave Propaga-ticn in Periodic Structures," J. Sound Vib.,

191. Rao, G.V. and Raju, K.K., "A Galerkin Finite 33, pp 223 236 (1973).Element Analysis of a Uniform Beam Carryinga Concentrated Mass and Rotary Inertia with 204. Sabir, A.B. and Ashwell, D.G., "A Comparisona Spring Hinge," J. Sound Vib., 37, pp 567.569 of Curved Beam Elements When Used in(1974). Vibration Problems," J. Sound Vib., 18,

pp 555.563 (1971).192. Rao, G.V., Sundarararamaiah, V., and Raju,

I.S., "Finite Element Analysis of Vibrations of 205. Seshadri, T.V., "Shock and Vibration AnalysisInitially Stressed Thin Shells of Revolution," Using Finite Element Techniques," Shock Vib.J. Sound Vib, 37 (1), pp 57.64 (1974). Dig,,7 (7), pp 75-95 (July 1975).

193. Rao, Y.V.K.S. and Sinha, P.K., "Vibrations of 206. Sheng-Taur-Mau, Pain, T.H., and Pin Tong,Sandwich Plates under Uniaxial Compression," "Vibration Analysis of Laminated Plates andAIAAJ., 12, p 1282 (1974). Shells by Hybrid Stress Elements," AIAA J.,

pp, pp 1450-1452 (1973).194. Raymond, D. Krieg and Samuel, W. Key,

"Transient Shell Response by Numerical Time 207. Shimizu, S., "Free Vibration Analysis ofIntegration," Intl. J. Numer. Methods Engr., Stiffened Plates," Second U.S.-Japan Seminar7, pp 273-286 (1973), on Matrix Methods of Structural Analysis and

Design, Univ. Alabama Press, pp 219-236195. Rehfield, L.W., "Higher Vibration Modes by (1972).

Matrix Iteration, ' J. Aircraft, 9, p 505 (1972).208. Shukut, T. and Ishihara, K., "The Analysis of

196. Rosen, Richard and Rubinstein, Moshe F., the Acoustic Field in Irregularly-shaped Rooms"Dynamic Analysis by Matrix Decomposition," by the Finite Element Method," J. Sound Vib.,Proc. ASCE, 94 (2), p 385 (1968). 29(1), pp 67-76 (1973).

197. Cook, Robert D, "Eigenvalue Problems with a 209. Singa Rao, K. and Amba Rao, C.L., "Vibrations'Mixed' Plate Element," AIAA J,, 7, p 982 of Beams with Overhangs," AIAA J., 11, pp(1969). 1445.1446 (1973).

198. Robinson, J, Structural Matrix Analysis for the 210. Singarao, K., Venkateswara Rao, G., and Raju,

Engineer, John Wiley and Sons (1966). I.S., "A Note on the Cylindrical Shell FiniteElement," Intl. J. Numer. Methods Engr.,

199. Ross, C TS., "Free Vibration of Thin Shells," 9, pp 245-250 (1975).J. Sound Vib ,39, pp 337-344 (1975).

211. Slyper, H.A., "Development of Explicit Stiff-200. Ross, C.T.F , "Finite Elements for the Vibra- ness and Mass Matrices for a Triangular Plate

tion of Cones and Cylinders," Intl J. Numer. Element," Intl. J. Solids Struc., 5, pp 241-249Methods Engr, 9, pp 833-845 (1975) (1969)

33

212. Klein, Stanley, "A Static and Dynamic Finite 223. Thomas, D.L., Wilson, J.M., and Wilson, R.R.,Element Shell Analysis with Experimental "Timoshenko Beam Finite Elements," J. SoundVerification," Intl. J. Numer. Methods Engr., Vib., 31, pp 315-330 (1973).3, pp 299-315 (197 1).

224. Thomas, J. and Dokumaci, E., "Simple Finite213. Stroebel, G.J., "Dynamic Coupling in an Un- Elements for Pre-twisted Blading Vibration,"

symmetrically Tapered Beam," J. Sound Vib., Aeron. Quart., 25, p 109 (1974).3 34, pp 275-283 (1974).

225. Thomas, J and Dokumaci, E., "Improved214. Subir, K. Sen and Phillip, L. Gould, "Free Finite Elements for Vibration Analysis of

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w P226. Turner, M.J., "Design of Minimum Mass Struc-iComputation of tures with Specified Natural Frequencies,"

Natural Frequencies of Structural Frames AIAA J.,5, pp 406.412,(1967).Using Exact Matrix Techniques," Theory andPractice in Finite Element Structural Analysis, 227. Valathur, M., Dove, R.C., and Albrecht, B.,Tokyo Univ. Press, pp 289-304 (1973). "Wave Propagation in a Thin Hollow Cone by

a Finite Element Method," J. Sound Vib.,216. Tabarrok, B., "A Variational Principle for the 24, pp 211-218 (1972).

Dynamic Analysit of Continua by a HybridFinite Element Method," Intl. J. Solids Struc,, 228. Venkateswara Rao, G., Venkataraman, J.,7 (3),pp 251-268 (1971). and Prakasa Rao, B., "Vibrations of Thick

Plates Using a High Precision Triangular Ele-217. Tabarrok, B. and Gass, N,, "Vibrations of ment," Nucl. Engr. Des., 31 (1), pp 102-105

Cylindrical Shells by Hybrid Finite Element (1974).Methods (SYM)," AIAA J., 10, p 1553 (1972).

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"A New Hybrid Cylindrical Shell Finite Ele- values in Some Vibration and Stability Pro-ment," J. Sound Vib , 21, pp 369-372 (1972) blems," Intl. J. Numer. Methods Engr., 5, pp

237-242 (1972).219. Tabarrok, B., Discussion of a paper by John

Robinson and M. Petyt, Intl. J. Numer. Meth- 230. Venkateswara Rao, G., Raju, I.S., and Ambaods Engr., 4, p 143 (1972). Rao, C.L., "Vibrations of Point Supported

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Plates Using a High Precision Triangular Ele-221 Thangam Babu, P.V. and Reddy, D.V., "Fre- ment," J Sound Vib., 34 (3), pp 444-445

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Straight Beam finite Element for Analysis Finite Element Method," Intl. J. Numer. Meth-of Vibrations of Curved Beams," J Sound Vib , ods Engr.,7, pp 185-194 (1973).26, pp 155-158 (1973).

34

233. Wang, C.K., Matrix Methods of Structural 244. Zienkiewicz, O.C., "The Finite Element Meth-Analysis, International Textbook (1966). od: From Intuition to Generality," Appl.

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Mechanics McGraw-Hill 1967).

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236. Wilson, R.R. and Brebbia, C.A., "Dynamic 247. Myklestad, N.O., "A New Method for Calcu-Behaviour of Steel Foundations for Turbo- lating Natural Modes of Uncoupled Bendingalternators," J. Sound Vib., a pp 405-416 Vibrations of Airplane Wings and Other Types(1971). of Beams," J. Aeron. Sci., 11, pp 153-162

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240. Yang, T.Y. and Kim, H., "Vibration and ference on Matrix Methods in StructuralBuckling of Shells Under Initial Stresses," MechanicsAIAA J., 11, pp 1525-1531 (1973).

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243. Yoshiaki Muda, "A New Relaxation Method forObtaining the Lowest Eigenvalue and Eigen-vector of a Matrix Equation," Intl. J. Numer,Methods Engr., 6, pp 511-529 (1973).

35

TABLE - Comparison of Some

Form of the Final Bound forBasis Matrix Equation Eigenvalue Remarks

Rayleigh-Ritz (R-R) (K] ¢j- A [M] 1 0 Upper Bound 1. Easy formulationMethod 2. Convenient for computer

programming

Modified Rayleigh- [F]tl- j[MJ]}=0 Upper bound 1. Requires concepts ofRitz Method [177] matrix force method

2. Best suited for one-dimensional problems

Galerkin Method [1761 [Ek ] r}- XEm] '}=0 Cannot be 1. Element degrees of free.predicted dom are more

2. Programming is similarto R-R method

3. Can consider nonlinearproblems for which aminimum energy prin-ciple may not exist

4. Can be extended toinitial value problems

Least-Square Method (P]jf}- A [Q1]t' Cannot be 1. Results in quadratic ei-[126+ predicted genvalue equations+ X,2IR] Jl4}= 0

2. Eigenvalues are generallycomplex

3. Programming complicat-ed

36

Finite Element Models

Form of the Final Bound forBasis Matrix Equation Elgenvalue Remarks

Pian's Type Hybrid [F (A)) f [M(,)J"} Cannot be 1. Results in transcendentalMethod [126] = 0 predicted matrix equation2. Conventional computer

programs cannot be useddirectly

3. Generally not a con-venient formulation

Collocation Method [D~ft - -tfl=0 Cannot be 1. Requires conceptsof[175] predicted the matrix force method

2. Best suited for one-dimensional problems

Prezemieniecki [A, X A2 -XA 3 . . .]{' Cannot be 1. Requires solutlon of aMethod (179] 0 predicted quadratic eigenvalue

problem2. Programming complicat-

ed3. Negative eiganvalues are

sometimes obtained

Stress Function [AX$ '- [13 J1 = 0 Cannot be 1. Involves suppression ofMethod [2191 predicted 'any frequency modes'

9, 37

BOOK REVIEWS

ELASTICITY IN ENGINEERING MECHANICSArthur P. Boresi ad Paul Lynn

Prentice-Hall, Inc. (1974)

This is an updated edition of an earlier book of the R* is unfortunate that the only method of solutionsame title published by Boresi in 1965. emphasized is the semi-inverse method. Students are

usually puzzled because direct methods are notChapter one is an introduction to elasticity and in- presented in elasticity. Some introduction to thecludes a description of boundary value problems. integral transform technique and complex variableChapter two deals with the theory of deformation methods should have been included. Nevertheless,and contains three appendices. Appendix 2A contains this text can be of great value in a first course inthe derivation of strain-displacement relations in elasticity, to both students and instructors, and it canorthogonal curvilinear coordinates. Appendix 2B be supplemented by handouts where desired.shows strain-displacement relations in general coordi-nates, and Appendix 2C is devoted to the derivationof displacement relations for special coordinates by Leon Y. Baharcartesian methods. Dept. of Mechanical Engineering and Mechanics

Drexel UniversityThe theory of stress is developed in Chapter 3. There Philadelphia, Pennsylvania 19104is an appendix on the application of the principleof virtual work. The two-dimensional theory ofelasticity in rectangular cartesian coordinates is thesubject of the fifth chapter. Appendix 5A describesthe effect of couple stresses on plane elasticity.

Two-dimensional elasticity in polar coordinates isdeveloped in Chapter 6. Appendix 6A gives theresults of stress concentration due to a circular holein a plate under the assumption of couple stresses.Chapter seven describes the solution to the problemof a prismatic bar subjected to end load.

An elementary treatment of the thermal stress prob-lem is found in Chapter 8. The final appendix re-capitulates the mathematics necessary to make thebook self-sufficient to the student.

The appendices to the individual chapters are themost refreshing part of the book. The derivations ofthe governing equations of elasticity in orthogonalcurvilinear coordinates are especially valuable becausethey are seldom found in introductory books. Thefact that the equations are derived vectorially makesthem all the more attractive

38

THE FINITE ELEMENT METHOD FOR ENGINEERSK. E. Huebner

John Wiley & Sons - New York (1975)I Finite element analysis now occupies a commanding The author applies various aspects of lubrication to

position in the areas of analysis and design. Huebner's gas bearings and journal bearings. He states that thebook is not elementary, as advertisements claim; isoparametric approach should have been consideredrather, it belongs at the intermediate level. The text is in fluid flow.divided into ten chapters and four appendices. Thebibliography is excellent,

In Chapter IX fluid mechanics problems are discussedin depth. A computer code with direct coding ofFundamental theory is the subject of the first chap- heat conduction problem is set up for the novice inter. The second chapter contains discussions of simple Chapter X.linear spring systems, one-dimensional heat flow,electrical networks, and the two-dimensional struc- The reviewer thinks that the book should be part oftural constant strain triangular element. In Chapter the library of the practicing engineer and scientistIII the author considers the composition of the involved in finite element applications. The bookfinite element and the equivalence between the finite should be published as two volumes, however, oneelement method and the Rayleigh.Ritz variational an elementary text, the other an advanced text.method. His formulation of the finite element meth- The section on structural dynamics should be ex-od is a good one.

panded to include component mode methods, sub-structuring, and transient response. The section onIn Chapter IV, which is the heart of the book, the gaussian quadrature applied t,) isoparametric elementsauthor describes simple triangular and quadrilateral and determination of transfer of stresses from the ielements, as well as the more complicated tetrahedral gaussian points to the nodes should also be expanded.and hexahedronal elements used in three.dimensionalstudies. There is discussion of two- and three-dimen.sional isoparametrIc elements and interpolation

Herb Saundersschemes; consideration of hexahedronal elements; General Electric Companyand derivation of linear, quadratic, and cubic iso- let o

parametric elements for both two and three dimen. Schenectady, New York 12345sions. The ex,,dnation is lucid and easy to follow.

The more complicated aspects of elasticity, includingthe role of variational methods, are found in the nextchapter. The displacement method and its applica-tions are considered. Fine mesh at a singularity, ihuplate bending problem, and a more detailed explana-tion of three-dimensional elements, including thehexahedronal element, conclude the chapter

Chapters VII and VIII consider such general problemsas heat conduction and inviscid incompressible andcompressible flow, Direct formulation of the finiteelements using variational methods are used Time-dependent fluid problems and transient motion uti-lizing recurrence relations are also described Thereviewer believes that the latter should have beenexpanded due to its importance in the dynamics ofreal structures

39

NEWS BRIEFSASTM PROPOSALS MEASURING IMPACT SOUND The session chairman is Paul M. Turkheimer, Director

TRANSMISSION OF FLOOR-CEILING of Program Development, Wyle Laboratories, ElASSEMBLIES NOW AVAILABLE Segundo, CA., Senior Member of the IES. The session

co-chairman is Thomas A. Colandrea, Director of theTwo proposals for methods to measure the impact Quality Assurance Div., General Atomic Co., Sansound transmission of floor-ceiling assemblies have Diego, CA, and chairman of the Nuclear Div., ASOC.been approved ty the American Society for Testing Frank Unmack, General Atomic Co., Staff Engineerand Materials (ASTM) Committee on Standards. responsible for equipment qualification, Fort St.

Vrain Project, completes the planning staff.The two proposals are: "Proposed Alternative Meth.od of Laboratory Measurement of Impact Sound The 9th Space Simulation Conference, being heldTransmission through Floor-Ceiling Assemblies Using at the same location from April 26 - 28, 1977, pro-a Modified Tapping Machine" and "Proposed Alter- vides an international forum for the discussion ofnative Method of Impact Sound Transmission space simulation and related topics. The technicalthrough Floor-Ceiling Assemblies Using a Live Walk- program is sponsored by the Institute of Environ-er." mental Sciences (ILS), American Institute of Aero-

nautics and Astronautics (AIAA), the AmericanAn ASTM Proposal is a specification, test method, Society for Testing and Materials (ASTM), and theclassification, definition, or recommended practice National Aeronautics and Space Administrationwhich has been approved by the sponsoring commit- (NASA). The ninth in this series of space simulationtee for publication as an "information only" docu- conferences is being hosted by the IES and will bement. A proposal is not submitted to Society letter conducted in conjunction with the IES 23rd Annualballot and is not a standard Copies of the two Technical Meeting and Equipment Exposition.proposals can be obtained from ASTM, 1916 RaceSt., Philadelphia, PA 19103. The Exhibits in the IES Equipment Exposition of

SPACE APPLICATIONS, ENERGY, TEST EQUIP-MENT, and INSTRUMENTATION in addition tothe Spacecraft and Test Facility Tours in the im-

INSTITUTE OF ENVIRONMENTAL SCIENCES mediate vi-inity are planned to enhance the technicalSCHEDULES SESSION ON NUCLEAR program.

AND FOSSIL FUEL ELECTRIC POWERGENERATION STATION QUALIFICATION For further information contact:

AT ITS 1977 ANNUAL MEFTINGB. L. Peterson, Executive Director

A special all day technical session entitled "Total Institute of Environmental SciencesQualification for Electric Power Generation Stations - 940 East Northwest HighwayNuclear and Fossil Fueled" will be held on 25 April Mount Prospect, Illinois 600561977, during the twenty-third Annual Meeting of Tele. (312) 255-1561the Institute of Environmental Sciences at the Mar-riott Hotel, Los Angeles, California. The speakersare prominent experts from government and industry,who will trace the requirements for qualifications ofsafety relmaed equipment from the NRC RegulatoryGuide through the QA plan to the completion of thepi ogram.

40

SHORT COURSES

FEBRUARY DIGITAL SIGNAL PROCESSINGDates: March 9- 11,1977

MODELING IN ENGINEERING DYNAMICS Place: Houston, TexasDates. February 28 - March 4, 1977 Objective: This seminar covers theory, operationPlace: San Antonio, Texas and applications -- plus additional capabilities suchObjective: This course is recommended for pro, as transient capture, amplitude probability, crossspective students who have a bachelor's degree in spectrum, cross correlation, convolution coherence,some field of engineering, physics, or mathematics. coherent output power, signal averaging and plentyThe intent of this course is to introduce and illustrate of demonstrations.to engineers, physicists, and scientists investigatingtransient phenomena the powerful tool of model Contact. Mr. Bob Kiefer, Spectral Dynamics Corp.analysis, and although the course is directed toward of San Diego, P.O. Box 671, San Diego, CA 92112experienced personnel, a newcomer to the fields of Tele. (714) 565-8211survivability analysis, terminal ballistics effects,safety engineering, engineering dynamics, etc., cankeep pace by diligently applying himself. DYNAMIC ANALYSIS OF STRUCTURES

Dates. March 14- 17, 1977Contact Mr Peter S. Westine, Southwest Research Place: Detroit, MichiganInstitute, P.O. Box 28510, San Antonio, TX 78284 Objective: This seminar provides practical laboratory

experience on getting good data and recognizing bad;diagnosing machinery with Vibration SpectrumAnalyzers, solving structural problems with TransferFunction Analyzers; and demonstrations using

MARCH state of the art FFT processors.

Contact. Mr. Bob Kieter, Spectral Dynamics Corp.CORRELATION AND COHERENCE ANALYSIS of San Diego, P.O. Box 671, San Diego, CA 92112FOR ACOUSTICS AND VIBRATION PROBLEMS Tele (714) 565-8211Dates. March 7 - 11, 1977Place UCLAObjective This course covers the latest practicaltechniques of correlation and coherence analysis-, MEASUREMENT SYSTEMS ENGINEERINGordinary,, multiple and partial -- for solving acoustics Dates: March 14- 19, 1977and vibration problems in physical systems Place. Phoenix, Arizona

Objective Program emphasis is on how to obtainContact Continuing Education in Engineering and valid cost-effective data in the field and in the labor-Mathematics, Short Courses, 6266 Boelter Hall, atory during the next decade through increased pro-UCLA Extension, Los Angeles, CA 90024 ductivity of data acquisition systems and groups.Tele (213) 825-1047 The latest developments in the new Unified Approach

to the Engineering of Measuring Systems to achievethese aims, will be presented.

Contact Prof P Stein, Short Course Director,5602 East Monte Rosa, Phoenix, AZ 85018Tele. (602) 945-4603/965-3124

41

APRIL MAY

INTRODUCTION TO VIBRATION AND SHOCK TURBOMACHINERY BLADING SEMINARTESTING, MEASUREMENT, ANALYSIS AND Dates May 3 -5,1977CALIBRATION Place. Rochester Institute of Technology,Dates: April 11 - 15,1977 Rochester, New YorkPlat:e. Boston, Massachusetts Objective: To introduce the vibration technologyObjective: Firms manufacturing weapons, aircraft, involved in the design and operation of turboma-missiles, naval or military vehicle systems or compo- chinery blades. Methods and instrumentation used tonents should consider sending their environmental measure and analyze blade vibration will be de-test personnel to this seminar. This course will con- scribed. Industrial experts and consultants willcentrate upon equipments and techniques rather than present theoretical background material and caseupon theory. histories. Panel sessions dealing with gas and steam

turbine blading problems and their solutions will

Contact Mr. W. Tustin, Tustin Institute of Tech- also be conducted.nology, Inc., 22 E. Los Olivos St., Santa Barbara,CA93105 Tele. (805) 963-1124 Contact: Dr. R. L. Eshleman, Director, Vibration

Institute, Suite 206, 101 W., 55th St., Clarendon Hills,IL 60514 Tele. (312) 654-2254/654-2053

RELIABILITY TESTING INSTITUTEDates: April 25 - 29, 1977Place. University of Arizona, TucsonObjective To provide Reliability Engineers, Product USNGEAssurance Engineers and Managers and all other en- Uaegineers and teachers with a working knowledge of Dates: May 16-June 16,1977analyzing component, equipment, and system per- Place Ashinto D.C.formance and failure data to determine the distri- Objective A sequence of three professional develop-butions of their times to failure, failure rates and ment courses will be presented to provide an under-

reliabilities; small sample size, short duration, Baye- standing of the technological content in general pur-

sian testing, suspended items testing; sequential pose finite element programs, and to provide training

testing; and others. in the use of NASTRAN. The courses and dates are

Contact. Dr. D. Kececioglu, Institute Director, Theory of Finite Elements, May 16 - 20, 1977

Aerospace and Mechanical Engrg. Dept., The Univ Static Structural Analysis Using NASTRAN,of Arizona, Bldg. 16, Tucson, AZ 85721Tele (602) 884-2495/884-3901/884-3054/884-1755

Dynamics and Nonlinear Structural AnalysisUsing NASTRAN, June 14 -27, 1977

Contact Dr H Schaeffer, Schaeffer Analysis,P.O. Box 761, Berwyn Station, College Park, MD20740 Tele. (301) 721-3788

42

ABSTRACTS FROMTHE CURRENT LITERATURElal rmteSI rthe Vibration Institute(eep

Copies of articles abstracted in the DIGEST are not available from the SVIC or(except

those generated by either organization). Inquiries should be directed to library resources. Government reports

can be obtained from the National Technical Information Service, Springfield, Va., 22151, by citing the AD-,

PB-, or N- number. Doctoral dissertations are available from University Microfilms (UM). 313 N. Fir St., Ann

Arbor, Mi. Addresses following the authors' names in the citation refer only to the first author. The list of

periodicals scanned by this journal is printed in issues 1, 6, and 12.

The proceedings of the "Conference on Vibrations in Rotating Machinery" held at the University of Cambridge,

England, September 15-17, 1976, are available from the Institute of Mechanical Engineers, 1 Birdcage Walk,

Westminster, London SW1 H9JJ England.

ABSTRACT CONTENTSANALYSIS AND DESIGN .... 44 Damping .............. 53 SYSTEMS ............... 69

Elastic ............... 54 69

Analogs and Analog Fatigue .............. 54 Absorber ..............

Computation ......... 44 Fluid ... ............ 54 Acoustic Isolation ....... 69

Analytical Methods ...... 44 Inelastic .............. 55 Noise Reduction ........ 69

Numerical Analysis ... .47 Viscoelastic ........... 55 Active Isolation ......... 70... . Aircraft .............. 70

Perturbation Methods .... 47 Bridges .............. 72

Stability Analysis ....... 47 EXPERIMENTATION ..... Bidg...............72

Variational Methods. . . .47 B u tin ..... 72

Finite Element Modeling. . 47 Balancing ............. 55 Construction .......... 73

Modeling ...... ...... 48 Diagnostics ............ 56 Earth ................ 73

Parameter Identification. . .48 Facilities ............. 56 Helicopters ............ 73

Design Information ..... .48 Instrumentation ........ 56 Human ............... 74

Criteria, Standards, and Techniques ............ 57 Isolation ............. 74

Material Handling ....... 75

Specifications ......... 48 Methal. kn and7

Surveys....... .. 48 COMPONENTS ......... 58 Mechanical............75

Mode Synthesis ..... 48 Forking and

Absorbers ............ 58 Forming............ 75

COMPUTER PROGRAMS .... 49 Beams, Strings, Rods .... 58 Off-Road Vehicles ....... 75

Bearings ............. 60 Pumps, Turbines, Fans,

General ....... .... 49 Blades . .... ........ 61 Compressors .......... 76

Columns ............ 62 Rail ................ 76

ENVIRONMENTS ......... 49 Cylinders.. ........ 63 Reactors .............

77

Gears ............... 63 Reciprocating Machines . . .78

Acoustic 49 Linkages ............. 63 Road ............... 79

Random . .... 50 Mechanical ......... .. 63 Rotors ....... ......... 82Satellite ..............

8

Seismic . ... . 50 Membranes, Films, and Sh i .............. 88

Shock . 50 Webs........... 64 Ship................

88

General Weapon . .51 Panels ............ 64 Spacecraft ........... 89

Pipes and Tubes ... ... 64 Structural ............ 90

P1IFNOMENOLO(Y ........ 52 Plates and Shells ....... 65 Turbomachinery ........ 90

Rings . . ..... 68 Useful Application ...... 91

Composite ., 52 Structural . 68

43

AIDthe lumped approximation of the distributed system. ThisANALYSIS AND DESIGN devlopen modifies, extends, and generalizes previous

studies that have been applied to a restricted class of systems.Throughout the article examples are given to introduce,motivate and illustrate the analysis.

ANALOGS AND ANALOGCOMPUTATION 77-234

Mode Theory of Wave Propagation in a BilineaMedium: The WKB ApproximationM. Hall

77-232 Navy Research Lab., Edgecliff 2027, Australia, J.The Dynamic Response of an Elastic Half Space with Acoust. Soc. Amer., 60 (4), pp 810-814 (Oct 1976)an Overlying Acoustic Fluide5 figs, 5 refsB.E. Bennett and G. HerrmannDept. of Applied Mechanics, Stanford Univ., Palo Kay Words: Waepropagation, EigenvalueproblemsAlto, CA., In: J. of Applied Mechanics, 98 (1), pp

39-42 (Mar 1976) Normal-mode calculations are made of wave propagation in abilinear medium with a free surface (the surface duct). The

AD 2 WKB approximation Is used. In the usual way, the Integralexpression for the field Is transformed Into an Infinite series

Key Words: Half.space, Elastic properties, Wave propaga- of the residues (normal modes) of the Integrand at Its polestion, Elastic waves, Seismic waves (elgenvalues). Approximations for the elgenvalues of the

trapped normal modes are obtained from WKB formulas. TheA class of dynamic problems involving a semi-infinite elastic elgenvalues of the untrapped modes are obtained by quadra-solid with an overlying semi-infinite acoustic fluid, subjected tic extrapolation of the elgenvalues of the trapped modes.at the plane Interface to arbitrary normal loading Is Inves. The actirscy of the real part of the elgenvalues of all modestigated. A method of solution Is proposed which reduces the Is Improved by taking Into account the Imaginary part whenclass of problems under study to that In which the fluid Is solving the characteristic equation.absent. A specific example is presented for an expandingdisk-shaped load Including numerical results for the sub-seismic range. 77-235

Difference Methods for Boundary Value Problemswith a Singularity of the First Kind

ANALYTICAL METHODS F.R. De Hoog and R. Weiss

Computer Centre, Australian National Univ., Can-

7berra 2600, Australia, SIAM J. Numer. Anal. 13 (5),~~~~77-233pp7 -70(t196

A Generalized Approach to the Solution of Variable p 753-760 (Oct 1976)

Systems Subjected to Arbitrary Source Functions and Key Words: Boundary value problems, Elgenvalue problemsBoundary ConditionsJ.C. Hassab The application of certain difference schemes (box, trape-

a Uzoidal, Euler and backward Euler) to the numerical solutionNaval und Sysem Cener, New pt of f.oundary value problems for nonlinear first order systemsS02840, J. Sound Vib., 48 (2), pp 277-291 (Sept of ordinary differential equations with a singularity of the1976) 15 refs first kind is examined, The solution of the linear elgenvalue

problem Is also considered.

Key Words: Layered materials, Forcing function

Layered inhomogeneous systems with or without forcing 77-236functions as represented by sources and boundary conditions The Minimum Ratio of Two Eigenvaluesare discussed. A common formulation to the constraints andphysical laws that apply is given by using the Fredholm inte- J.B. Keller

gral equation. A unified and versatile solution is developed Courant Inst. of Mathematical Sciences, New Yorkanalytically to treat these problems under forced or un- Univ, New York, NY 10012, SIAM J. Appl. Math.,forced conditions. Then the approach is dissected to imple- 31 (3), pp 485-491 (Nov 1976) 3 figs, 1 refment the results on the computer or to treat, at the outset, Sponsored by ONR

4 44

Key Words: Eigenvalue problems G.G. Kleinstein and M.D. GunzburgerJoint Institute for Acoustics and Flight Sciences,

The first two eigenvalues, X, and ,2, of the problem y" + The George Washington Univ., Hampton, VA 23665,)(x)y - 0, y(±%) - 0 re considered. The method of analysis J. Sound Vib., 48 (2), pp 169-178 (Sept 1976)is applicable to other similar problems with inequtaltty 3 figs, 4 refsconstraints.

Sponsored by NASA

Key Words: Frequencies, Elastic waves, Wave number

Analytical Theory of Nonlinear Oeilations IV: The An integral conservation law for wave numbers is considered.Periodic Oscillations of the Equation X - e(]-x2n+ 2 ) In order to test the validity of the proposed conservationx + x~h+l = Ca cos wt, a>O. w>O Independent of e law, a complete solution for the reflection and transmission

of an acoustic wave impinging normally on a materiel inter-face moving at a constant speed is derived. Results are stated

Dept. of Mathematics, Univ. of Lagos, Lagos, Nigeria, concerning frequency and wave number relations across a

SIAM J. Appl. Math., 31 (2), pp 345-357 (Sept 1976) shock front as predicted by the proposed conservation law.11 refs

Key Words: Nonlinear response, Periodic response 77.240

A form of the well-known equation of Van der Pol with a Exact Solution for Guided Sound Transiasionforcing term is U - - x2) + x - ea cos Cut. In this paper Through a Double Partitiona generalized version of Van der Pal's equation Is introduced N. Romilly

- e(1 - x2n+2)j + x2 n+1 = ea cos (,, where QV1 is an Dept. of Applied Mathematical Studies, Univ. ofinteger and &>0, W0 are Independent of e. A preliminary Leeds, Leeds LS2 9JT, England, J. Sound Vib.,study of this equation for a general n and small e is made.In addition a more detailed study of its special case (n"l and 48 (2), pp 243-249 (Sept 1976) 5 refse is small) is considered. Results on the exact number of itsperiodic oscillations, the relations between the stationary Key Words: Sound transmission, Wells, Membranesvalues (amplitudes), least periods of its periodic oscillations,and the ways some of the periodic oscillations appear or The exact solution is obtained for the problem of the trana.disappear as the parameters a, (Ovary are given, mission of a symmetric sound wave through a double parti-

tion contained in a parallel-plane wveguide with rigidwalls. A membrane model Is used for the two leaves of thepartition. The solution involves a number of infinite series

77.238 which converge rapidly. No modifications are necessary for

Analytical Theory of Nonlinear Oscillations. i: the region of the critical frequencies. Full details re given forSmall Periodic Osillations of Equations of the the particular case of an incident plane wave. The result,

which is suitable for computation, Involves four convergingSecond Or r infinite series.C Obi

Dept of Mathematics, Univ. of Lagos, Lagos, Nigeria,SIAM J. Appl. Math., 31 (2), pp 334-344 (Sept 1976) 77.2416 refs

Spatial Cros-Correlation of Acoustic Pressres in

Key Words: Nonlinear response, Periodic response Steady and Decaying Reverberant Sound FieldsC.F. Chien and W.W. Soroka

The equations considered are perturbations of the equation Dept. of Mechanical Engrg., Univ. of California,iJ + g(x) = 0. The results established are a number of theo- Berkeley, CA 94720, J. Sound Vib., 48 (2), pp 235-rems on the existence and number of those periodic oscil. 242 (Sept 1976) 10 refslations (of the equations considered) which themselves areperturbations of the constant solutions of R + g(x) = 0. Thetheorems are stated in terms of small periodic oscillations. Key Words: Cross-correlation technique, Sound pressures,

Reverberation chambers

77.239

Frequency Modulation at a Moving Material Interface

and a Conservation Law for Wave Number

45

From the equation for the steedy state sound pressure distri- 77-244bution produced in a rectangular reverberation chamber by a Narrow-Band Excitation of a Nonlinear Oscillatorpoint source, and by using the usual high frequency Ioxi- W.C. Lennox and Y.C. Kuakmtions, the cron-corrotion function for two points not toof ons er rradou soureon function for twopointsWnt th Dept. of Civil Engrg., Univ. of Waterloo, Waterloo,far apart for a random source position is discussed. When the (ue17)4fg,6rfsame approach is used on the equation for sound pressure Ontario, Canada, J. Appl. Mech., Trans. ASME, 43 (2),decay when the point surce excitation is cut off, thi Crot - pp 340-344 (June 1976) 4 figs, 6 refscorrelation function obtained for the initial portion of thedec corresponds with that determined experimentally by Key Words: Oscillators, Narrow-band excitationBlachandren and Robinson.

A nonlinear oscillator characterized by a hardening-typerestoring force Is excited by stationary narrow-bend nolse. Theanalysis Is based on the concept of quasi-static amplitude and

77-42 phase values which is ex, tly opposite to the stochastic orA Comparison of Three Perturbation Methods for Markov approach with its associated Fokker-Planck equation.Non-Linear Hyperbolic Waves The approach, in effect, replaces a systern with memory with

A.H. Nayfeh and A. Kluwick one without memory. The existence of jumps Is demonstrated

Dept. of Engrg. Science and Mechanics, Virginia and an expression for the probability of the occurrence of

Polytechnic Inst. and State Univ., Blacksburg, VA, jumps is derived.

J. Sound Vib., 48 (2), pp 293-299 (Sept 1976) 7 refs

Key Words: Perturbation theory, Wave propagation 77-245The Forced Vibration of Singly Modified Damped

Simple wave solutions of non-linear hyperbolic equations are Eaitic Surface Sy temsstudied by using the method of renormalization, the analytic R.G. Jacquotmethod of characteristics, and the method of multiple sceles.It Is shown that the results ot the method of renormalization Dept. of Electrical Engro., Univ. of Wyoming, Laramie,depend on whether the potential function or the velocity is WY 82070, J. Sound Vib., 48 (2), pp 195-201 (Septnormalized. 1976) 2 figs, 15 refs

Key Words: Lumped parameter method, Natural frequencies,

77-243 Mode shapes, Forced vibration, Plates, Shells, Membranes,

Perturbed Nonlinear Oscillations Beams, Viscous damping

B. Kaper A technique Is developed to predict the forced vibration ofDept. of Mathematics, Univ. of Groningen, Groningen, membranes, beams, plates or shells when they have attachedThe Netherlands, SIAM J. Appl. Math., 31 (3), pp to them at a single point a linear lumped parameter element or519-546 (Nov 1976) 9 refs assembly of elements, The distributed parameter element Is

treated as viscously damped and the lumped parameter assem-

bly may also contain viscous dampers. Solution Is obtainedKey Words: Oscillat on, Nonlinear response, Perturbation in terms of generalized Fourier series in the unmodified sigen-theory functions for the distributed portion of the system and the

principle of superposition is used to handle the ImposedFor a class of nonlinear oscillation problems described by a forces and those generated at the attachment. The methodsecond order ordinary differential equation containing a small is illustrated by investigating a uniformly forced simply sup-nonnegtive perturbation parameter; asymptotic solutions ported rectangular plate with a lumped mass at Its centerwhich are uniformly valid as approaches zero are determined and that of a point forced simple beam with a rigid pin supporton intervals of the order of magnitude one over the perturba- imposed at some arbitrary point.tion parameter. These asymptotic solutions are obtained fromformal asymptotic solutions subject to a condition in orderthat they approximate the exact solution asymptotically.These formal asymptotic solutions are constructed in the formof a finite generalized asymptotic power series involving 77-246functions of the periodic two variable type. Periodic Solutions of Second Order Differential Equa-

tions with Nonlinear, Nondifferentiable Damping

46

W.R. Utz Key Words: Stability analysis, Multidegree of freedomDept. of Mathematics, Univ. of Missouri-Columbia, systems, Parametric excitationColumbia, MO 65201, SIAM J. App). Math., 31 (3), The existence of three different types of unstable region inpp 504-510 (Nov 1976) 11 refs mulit-degree of freedom linear systems undergoing beating

two-frequency parametric loading is demonstrated. Two andKey Words: Equations of motion five degree of freedom digital system models exhibiting

simple and combination resonant response to quasi-periodicFor the equation x" + O(x')+f(x)-O, periodic solutions are parametric loading are discussed. Two degree of freedomestablished without a requirement of differentiability of 0. mathematical models sbjected to beating Input situationsBy separate methods, 0 is treated as even and then odd. In are analyzed. Analytical, numerical, electronic analog andthe odd case, a comparison theorem is provided when f(x) experimental studies are described and the practicalitles and

2w x. relevance of the methods used are Indicated.

VARIATIONAL METHODSNUMERICAL ANALYSIS(See Nos. 319,429)

77.249A Reduction Scheme for Problems of Structural

PERTURBATION METHODS Dynmic.i' (See Nos. 262, 421)e o ,T.J.R. Hughes, H.M. Hilber and R.L. Taylor

Div. of Structural Engrg. and Structural Mech., Dept.of Civil Engrg., Univ. of California, Berkeley, CA.,

STABILITY ANALYSIS Intl. J. Solids Struc., 12 (11), pp 749-767 (1976)(Also see Nos. 408,427) 6 figs, 14 refs

Key Words: Finite element technique, Variational methods,77-247 Dynamic structural analysisStability of Nonlinear Osclatory SystemsJ. Lin A method for reducing the size of finite element systems inPh.D. Thesis, State Univ. of New York at Stony dynamics is presented. The technique Is based upon a varia-

tional theorem in which it Is admissible to describe theBrook, 203 pp (1976) inertial properties of structures by way of independentUM 76-19, 691 displacement, velocity and momentum fields. This theorem

allows us to construct reduced systems for problems InKey Words: Stability methods, Oscillators structural mechanics which retain the full rate of convergence

of systems employing "consistent" mass matrices. An errorThe averaging technique of Krylov.Bogoliubov.Mlkopolskii analysis of the scheme and numerical examples are presented.(K-B-M) to solve several nonlinear models with oscillatorybehavior used to study a class of one predator - one preysystems which incorporate a functional response of the FINITE ELEMENT MODELINGpredator. The K-B-M method allows us to calculate, under (Also see No. 249)certain conditions, the radius of the limit cycle, and therenormalized frequency of the system's response.

77-250Analysis of Elastic.Plastic Impact Involving Severe

77.248 DistortionsParametric Instability Regions in Multi-Degree of G R. JohnsonFreedom Systems Under Quasi-Periodic Beating Government and Aeronautical Products Div., Honey-Input Excitation well, Inc., Minneapolis, MN, J. Appl. Mech., Trans.D C McWhannell ASME, 43 (3), pp 439-444 (Sept 1976) 5 figs, 12 refsDept. of Mech Engrg., Univ. of Southampton, Sou=th-ampton S09 5NH, England, J. Sound Vib , 48 (1), Key Words: Impact response, Finite element technique,pp 73-81 (Sept 8, 1976) 2 figs, 4 refs Computer programs

47

A Lagrangian analysis technique is presented for two-dimen- V.J. Conferslional axl-symmetric impact problems involving elastic- Human Engrg. Lab., Aberdeen Proving Ground, MD,plastic flow. This technique is based on a triangular finite- 41 pp (July 1976) (Supersedes rept. dated Sept 1971,element formulation rather than the quadrilateral formula Ation generally used In comparable finite.difference methods. AD-731,468)The formulation of the technique and illustrative examples AD-A028 277/2GAare included.

Key Words: Noise, Shock, Human response, Bibliographies

MODELING The bibliography Is an annotated compilation of 95 formallypublished reports dealing with noise and blest. The researchwas performed by personnel of the U.S. Army Human

77-251 Engineering Laboratory and covers the period from 1956

Modal Simulation of the Intennittent Vibration of through July 1976.

TurborotoraB. GrabowskiVDI Fortschrittberichte (Progress Reports of the MODE SYNTHESISVDI), Series 11, (25), 122 pp (1976) 45 figs, Avail: (Alsosee No. 261)

VDI Verlag, GmbH, 4 Ddsseldorf 1, Postfach 1139,Fed. Rep. of Germany. Summarized in VDI-Z,118(17/18) (ept 1976) - 77.253(In German) Computed Restitution of Structural Natural Modesfrom Inappropriate Excitations

Key Words: Rotors, Critical speeds, Modal models X.T. NguyenOffice National d'Etudes at de Recherches Aero.

Two different variations of the modal simulation of turbo- spatiales, Paris, France, Rept. No. ONERA-NT-rotors are Investigated. In one, the real elgenshapes of the 1975-9, 54 pp (1975)mechanical system are used for the transformation; in the (in French;, English summary)other the orthogonal vibration planes of an uncoupled N7622597mechanical system are used.

Key Words: Modal synthesisPARAMETER IDENTIFICATION

(See Nos. 349, 373. 374) After recalling the theoretical bases and experimental me.thods used for ground vibration tests, the fundamentalproblem of modal data determination Is formulated. Theproposed method makes It possible to determine the vibra.

DESIGN INFORMATION tory characteristics of a linear, weakly damped struc'ure(See No. 428) from scannings around the resonance frequency of each

mode with different configurations of excitation forces.

CRITERIA, STANDARDS, ANDSPECIFICATIONS 77.254

(See Nos. 347, 372) A Stationarity Principle for the Eigenvalue Problemfor Rotating StructuresL. Meirovitch

SURVEYS Virginia Polytechnic Institute and State Univ.,(Also see No. 259) Blacksburg, VA, AIAA J., 14 (10), pp 1387-1394

(Oct 1976) 1 fig, 15 refs

77-252 Key Words: Eigenvalue problems, Rotating structures,

Noise and Blast (1956-1976). An Annotated Biblio- Gyroscopes, Spacecraftgraphy of Research Performed at the Human Engi.neering Laboratory

48

An important question associated with the elgenvalue prob- J.R. Houghtonlam for flexible gyroscopic systems is that of discretization Ph.D. Thesis, Vanderbilt Univ., 200 pp (1976)of continuous elastic members. If discretization is performed UM 76-22, 345by the assumed modes method, the question arises as to thetype of functions to be used in series expansions. In particu-lar, the question is whether one should use rotating-append- Key Words: Acoustic signatures, Spectrum analysisage eigenfunctions, fixed-based elgenfunctions, or any otherset of admissible functions. A method for the signature analysis of pulses in the fre-

quency domain and the time domain is presented. Fourierspectrum, Fourier transfer function, shock spectrum andshock spectrum ratio were examined in the frequency do-main analysis end pulse shape deconvolution was developedCOM PUTER PROGRAM S for use In the time domain analysis. Comparisons of therelative performance of each analysis technique are made forthe characterization of acoustic emission pulses recorded bya measuring system. To demonstrate the relative sensltivityof each of the methods to small changes in the pulse shape,signatures of computer modeled systems with analytical

GENERAL pulses are presented. Optimization techniques are developed(Also see Nos. 281, 322, 337, 354, 4401 and used to indicate the best design parameters values for

deconvolution of the pulse shape. Several experiments arepresented that test the pulse signature analysis methods ondifferent acoustic emission sources.

77.255Dynamics of Inertia Variant MachineryGOC. HudPh.D. Thesis, Univ. of Pennsylvania, 247 pp (1976) 77-257UM 76-22,706 Sound Production by Organ Flue Pipes

N.H. FletcherKey Words: Computer programs, Equations of motion, Dept. of Physics, Univ. of New England, Armidale,Mechanical systems New South Wales 2351, Australia, J. Acoust. Soc.

Amer , 60 (4), pp 926-936 (Oct 1976) 6 figs, 25 refsIt is shown in this dissertation how Lagrange's form ofd'Alembert's Principle can be successfully adapted to asystematic modeling procedure for automatically generating Key Words: Sound generation, Musical Instrumentsthe equations of motion, and associated equations of con- A nonparametric treatment Is devised to allow calculation ofstraint, for multi-loop, multi-freedom mechanical systems. the transient and steady-state acoustic spectra of a flue organUsing this principle, a general purpose, user-oriented com- pipe, given the geometry of Its construction and the pressureputer program, DYMAC, for solving planar problems in- variation in Its foot, Explicit formulation is given for thevolving either lower or higher order pairings has been devel- fundamental and the first and second upper partils. Aoped and tested. scaling law is set out, relating the behavior of pipes of differ-

ent fundamental frequencies, and several illustrative calcula-tions are performed.

ENVIRONMENTS 7-5Second Harmonic Generation in Elastic Surface

Waves on an Isotropic SolidG.V. Anand

ACOUSTIC Dept. of Electrical Communication Engrg., Indian(Also see Nos 240, 241, 252, 291, 292, 351, 354, Inst of Science, Bangalore-560012, India, Intl, J.

371,376,387,388) Nonlinear Mech., 11 (4), pp 277-284 (1976) 15 refs

Key Words: Elastic waves, Wave propagation, Isotropy

77-256 A procedure for solving the problem of nonlinear propaga-Pulse Analysis of Acoustic Emission Signals tion of elastic surface waves is given. An expression for the

particle displacement of the second harmonic is obtained.

A 49

M, 7i NMVt4

77-259 refining existing schemes for the synthesis of waveforms.Urban Noise Pollution (A Bibliography with Ab. The contribution from very low frequencies to a modalstlracts) synthesis of an acoustic-gravity waveform is clarified. A guide

is provided for adapting a computer program to include suchE.J. Lehmann contributions in the synthesis of waveforms. A geometricNational Technical Information Service, Springfield, acoustical scheme is outlined for the prediction of the am-VA., 109 pp (July 1976) (Supersedes NTIS-PS- plitudes of waves that propagate over long distances. A75/544 and NTIS/PS-74/106) number of FORTRAN subprograms are provided that exem-

NTIS/PS-76/0585/OGA plify the numerical implementation of this scheme. Recom-mendations are given for the refinement at low and highfrequencies of schemes for the synthesis of waveforms.

Key Words: Urban noise, Traffic noise, Noise reduction,Bibliographies

RANDOMAspects of noise in the urban environment are covered. The (See No, 376)topics were chosen for their interest to urban planners andInclude citizen attitudes, transportation noise, and noiseabatement techniques.

SEISMIC(Also see Nos. 331,442)

77-260Measurement and Prediction of Sound Attenuation 77.262by Buildings Using Acoustic Modeling Techniques A Perturbation Scheme for Obtaining Partial Deriva-E.S. Ivey tives of Love-Wave Group-Velocity DispersionPh.D. Thesis, Univ of Massachusetts, 117 pp (1976) D. KosloffUM 76-22, 266 Seismological Lab., California Inst. of Technology,,

Pasadena, CA, Rept. No. AFOSR-TR.76-0821, 10 ppKey Words: Urban noise, Noise reduction, Buildings, Mathe- (Feb 10, 1975) (Published in the Bull, of the Seismo-matical models logical Society of America, 65 (6), pp 1753-1760

The attenuation of sound as it propagates over building-size (Dec 1975))barriers is an important part of the general urban and subur- AD-A027 989/3GAban noise propagation problem. While the sound attenuationover finite thickness barriers has been studied analytically Key Words: Seismic waves, Perturbation theory, Computerand experimentally, these studies have not produced a con. programsvenlent methodology for predicting the noise attenuationcharacteristics of building-size barriers. These studies have, A method is derived for obtaining partial derivatives ofhowever, identified several factors which influence the Love-wave group-velocity spectra for a layered medium usingnoise attenuation of any particular barrier configuration. a second-order perturbation theory. These partials are a

prerequisite for systematic inversion of group-velocityspectra. Mathematically the equation of motion and bound-

77-261 ary conditions for Love waves are a singular Sturm Liouville77261type eigenvalue problem.Mathematical Models of Acoustic and Acoustic-

Gravity Wave Propagation in Fluids with Height-Dependent Sound Velocities SHOCKW A. Kinney (Also see Nos. 250, 252, 293, 360, 363, 365, 380,

W . iney381,382,383, 391, 392,393, 394, 399, 435, 436)Ph.D. Thesis, Georgia Inst of Technology, 172 pp

(1976)UM 76-23,735 77-263

A Computer-Oriented Deterministic-Cum-Probabilis-Key Words: Elastic waves, Waves propagation, Modal syn- tic Approach for the Extreme Load Design of Coin-thesis, Mathematical models, Computer programs tiexAStrcur

plex Structures

Several problems which relate to the propagation of acoustic H Kamiland acoustic-gravity waves in a medium whose properties Engineering Decision Analysis Co., Inc , 480 Califor-vary with height only are considered with the intent of ma Ave.,, Palo Alto, CA, Computers and Stiuc, 6

(4/5), pp 375-379 (Aug/Oct 1976) 16 refs

50

Key Words: Computer aided design. Shock-resistant design, P.J. Chen, M.F. McCarthy and T.R. O'LearyInteraction: structure-foundation, Probability theory Sandia Labs., Albuquerque, NM, Archive Rational

Mech. Anal., 62 (2), pp 189-207 (Aug 18, 1976)To obtain a rational, economical, and reliable structuraldesign under extreme loading conditions, a combination of 12 refs

deterministic and probabilistic approaches need t be usd. Sponsored by ERDAComputers can be effectively used to combine these twoapproaches together and apply them to the design of struc- Key Words: Shock wave propagationtures. State-of-the-art techniques are reviewed for the deter- The behavior of one-dimensional shock and accelerationministic and the probabilistic methods. Computers can be waves in deformable dielectric materials with memory anused for the application of a combined deterministic-cum, exemined. The differential equation which the amplitude ofprobabilistic approach in the following two areas: (1) deter- the shock must obey was derived. The properties of the

- mination of design loads for multiple load cases by con- electric field during shock transition are examined andbining the individual load cases probabilistically outside certain of Its properties in terms of the material propertiesthe structure-foundation system and then determining we derived. The differential equation which the amplitude ofthe element load cas deterministically inside the system; the acceleration wave must satisfy is derived.(2) determination of the reliability of the foundation-struc-ture system by first modeling the member property variationsat element levol and then generating the system reliabilityusing matrix algebra techniques, along with the system stiff- GENERAL WEAPONness matrix. Methodologies are presented to describe how the (Also see No. 285)above objectives can be achieved. Limitations of the pro-posed methodologies are described and recommendationsfor future studies are made.

77.266Underground Ammunition Storage. Report 1. Teat

77-264 Program, Instrumentation, and Data ReductionCorelo oA. Skjeltorp, T. Hegdahl and A. JenssenCorrelation of Impact and Explovely Created Office of Test and Development, Norwegian DefenceGround Shock Phenomena Construction Service, Oslo, Norway, Rept. No. Forti-M B. Ford fikatorisk Notat-80172, 68 pp (Sept 1975) (See alsoArmy Engineer Waterways Experiment Station, AD-A027 064)

Vicksburg, MS, Rept. No. WES-MP-N-76-10, 74 pp AD-A027 063/7GA

(June 1976)

AD-A027 059/5GA Key Words: Underground structures, Tunnels. Ammunition,Storage, Explosion effects, Model testing

Key Words: Ground motion, Explosion effects This report Is the first in a series of five describing an exten-sive series of model tests on air blast propagation in under-

A technique for prediction of ground motions In the transi- ground ammunition storage sites. A general description istion and far-out regions where outrunning surface waves are given of the scope and purpose of the tests as well as atransporting most of the energy is described. A statistically presentation of the principles of scaling and various intrinsicsignificant number of surface ground motion velocity mea- uncertainties which may cause problems in the modelingsurements were produced by detonating high-explosive (HE) technique. The report also surveys the test program, instru-spheres of differing yields and depths of burst. Similar mentation, and data reduction for the whole test series.measurements were obtained from energy sources realizedby impacting free-falling weights. The tests in this study wereconducted on a nearly homogeneous sandstone formation inOctober 1973 at a site near Grand Junction, Colorado, along 77-267with Project CENSE (Coupling Efficiency of Near Surface Underground Ammunition Storage. Blast PropagationExplosions). in the Tunnel System. Report 11. A. Chamber Pres-

sureA Skjeltorp,T. Hegdahl and A. Jenssen

77-265 Office of Test and Development, Norwegian DefenceOne-Dimensional Shock and Acceleration Waves in Construction Serivce, Oslo, Norway, Rept. No. Forti-Deformable Dielectric Materials with Memory fikatorisk Notat-79/72, 42 pp (Sept 1975) (See also

AD-A027 065)

AD-A027 064/5GA

51

Key Words: Underground structures, Tunnels, Ammunition, Key Words: Underground structures, Tunnels. Ammunition,Storage, Explosion effects, Shock wave propagation Storage, Blast effects, Shock wave propagation, Model

testingThe pressure-time history for detonation and burning ofvarious explosives in confined regions (chambers) is exam- Model tests have been performed to determine the blasttried. In the detonation tests, a systematic study was per- wave propagation in the tunnel system of undergroundformed for three types of explosives (TNT, PETN, and ammunition storage sites with connected storage chambers.Dynamite) for different loading densities and chamber A total of 18 different configurations were tested withventing. Four additional explosives (AN/FO, ROX, ALUMIT, variable chamber volumes (300 - 15200 cc), angles betweenand COMP.8) were also tested over a limited range of loading the branch and main passageway (35 - 90 deg.). and ratiosdensities in a dosed chamber. To determine the maximum between the cross sections of the branch and main pssae-'chamber pressure', carefully designed electrical filters were way (0.125 - 0.5). The models were In linear scales 1:40 toemployed to remove the high frequency ringing in the 1:100 of typical full scale installations. By analyzing thepreumre-time recordings. Some preliminary tests were per. data, it was possible to obtain relatively simple scaling rela-formed to determine the ignitability for the explosives used tionships which contain all of the most important geometri-in the detonation tests using the closed bomb method. cal parameters. The model data are compared with one large

scale test and fair agreement is found.

77-268Underground Ammunition Storage. Blast Propagation 77-270in the Tunnel System. Report Ill. A. Single Chamber Underground Ammunition Storage. Blast PropagationStorage. Variable Tunnel Diameter and Variable in the Tunnel System. Report V. A. ConnectedChamber Volume Chamber Storage Blast Load on Doors in Three SitesA. Skjeltorp. T. Hegdahl and A. Jenssen A. Skjeltorp, T. Hegdahl and A. JenssenOffice of Test and Development, Norwegian Defence Office of Test and Development, Norwegian DefenceConstruction Service, Oslo, Norway, Rept. No. Forti- Construction Service, Oslo, Norway, Rept. No. Forti-fikatorisk Notat-81/72, 60 pp (June 1975) (See also fikatorisk Notat-83172, 32 pp (Sept 1975) (See alsoAD-A027 066) AD-027 063)AD-A027 065/2GA AD-A027 067/8GA

Key Words: Underground structures, Tunnels, Ammunition, Key Words: Underground structures, Tunnels, Ammunition,Storage, Blast effects, Shock wave propagation, Model Storage, Blest effects, Shock wave propagation, Modeltesting testing

Model tests have been performed to determine the blast The result from airblast measurements in models of threewave propagation in the tunnel system of underground single particular types of underground ammunition storage sites Ischamber storage sites. A total of nine configurations were reviewed. The use of simple scaling relationship are shown totested with variable tunnel diameters (6.7 - 13.5 cm), cham- reproduce the blast data reasonably well for a wide range ofber volumes (7250 - 15200 cc), and TNT loading densities geometrical parameters.(1 - 70 kg/cu m). Comparisons are given with one large scaletest and the Importance of well roughness attenuation Isdiscussed. PHENOMENOLOGY77.269Underground Ammunition Storage. Blast Propagationin the Tunnel System. Report IV. A. Connected COMPOSITEChamber Storage. Variable Chamber Volume and (Also see Nos. 283, 297, 340)Variable Angle between Branch and Main PassagewayA Skjeltorp, T. Hegdahl and A. JenssenOffice of Test and Development, Norwegian DefenceConstruction Service, Oslo, Norway, Rept. No. Forti- 77-271fikatorisk Notat-82/72, 48 pp (Nov 1975) (See also Harmonic Waves in Three-Dimensional Elastic Com-AD-027 067) positesAD-A027 066/OGA

52

S. Minagawa and S. Nemat-Nasser 77-274Dept. of Civil Engrg., Northwestern Univ., Evanston, Critical Factors for Freque -cy.Dependent FatigueIL 60201, Intl. J. Solids Struc., 12 (11), pp 769- ProcessesinCompositeMaterdals777 (1976) 7 figs, 9 refs W.W. Stinchcomb, K.L. Reifsnider and R.S. Williams

Engrg. Science and Mechanics Dept., Virginia Poly-Key Words: Composite materials, Wave propagation technic Inst. and State Univ., Blacksburg, VA 24061,

Exptl. Mech., 16 (9), pp 343.348 (Sept 1976) 6 figs,For a periodic elastic composite which consists of a matrix 7 refsand fibers with finite dimensions (i.e., a three-dimensional 17 re

problem), estimates for elgenfrequencles and eigenfunctions Sponsored by AFOSRare given. Calculations are based on a new quotient proposedby Nemat-Nasser. The periodic character of the eigenfre. Key Words: Composite materials, Fatigue (materials),quencles is pointed out, and illustrative examples are given. Frequency response

Several fatigue-test parameters, including cyclic frequency,prfltigue material conditioning (prelooding and step load.

77.272 Ing) and tet-control modes (strain control and load control)Analyoa of Wave Mode in Compound Elasic Circular are investigated and their effect on the fatigue response ofRods composite materials Is discuseed. A conceptual model booed

on the test results is offered to aid in the understanding ofH. Toda and H Fukuoka fatigue procese in composite materials and the effect ofDept. of Engrg. Science, Osaka Univ., Toyonaka, frequency on fatigue response.Japan, Bull. JSME, 19 (133), pp 755-760 (July 1976)5 figs, 7 refs DAMPING

Key Words: Rods, Composite materials, Frequency equation

An analysis of the wave modes In compound elastic circular 77-275rods, which consists of a circular matrix rod and a circularheaeth tube made of different materials, was executed. DampiTn Synthesis from Substructure Tests: T.K. Hasselman

J.H. Wiggins Company, Redondo Beach, CA, AIAA

77-273 J., 14 (10), pp 1409-1418 (Oct 1976) 6 figs, 10 refs

On Continuum Modeling of the Dynamic Behavior Key Words: Modal dlmpingof Layered CompositesG. Herrmann, R.K. Kaul and T.J. Delph A new method Is proposed for synthesizing structural damp.Dept. of Mech. Engrg., Div. AppI. Mech., Stanford ing from substructure test data. It utilizes the off-diagonalUniv., Stanford, CA 94305, Archives of Mechanics, coupling terms In the substructure modal damping matricesas well a the diagonal terms which correspond to uncoupled28 (3), pp 405421 (1976) 17 figs, 33 refs modal damping. The coupling terms are evaluated from com-Sponsored by AFOSR plex resonant response measured at each substructure mode.

Tentative accuracy requirements on experimental data andKey Words:, Composite materials, Continuum mechanics, date reduction are suggested.Mathematical models

A hierarchy of new approximate continuum theories tomodel the dynamic behavior of layered composites is ad- 77.276vanced. In each approximation the approximate frequency Steady Impact Vibration of a Body Having Hysteresispectra are matched as closely as possible with the exact Colision Characteristics (2nd Report, System withspectra, and the range of validity of each approximation is Quadrilateral Hysteresis Loop Characteristics forascertained. The procedure followed is like that employed Fore of Restitution)in deriving approximate plate theories. Similarities to certainphenomena in solid state physics are pointed out, and an . Maezawa and T. Watanabeanalogy to a system with one degree of freedom is drawn. Dept. of Engrg., Yamanashi Univ., Kofu, Japan,

Bull. JSME, 19 (134), pp 902-911 (Aug 1976)9 figs, 6 refs

53

Key Words: Hysteretic damping, Harmonic excitation FATIGUEt(See No. 274)

Steady impact vibrations in a mechanical system subjectedto various harmonic exciting forces are analyzed by a perfectFourier series method reinforced by convergent improvementby means of a series transformation. The hysteresis loop FLUIDcharacterstins for force of restitution are assumed to be (Also see No. 331)composed of four straight-1ine segments.

77-279ELASTIC The Added Mass and Damping Coefficients of and

the Excitation Forces on Four Axisymnnetric OceanPlatforms

77-277 K.J. BaiVibrations of Solids in Term. of Quantum Eqpations Ship Performance Dept., David W. Taylor Naval Shipof Elasticity Res. and Dev. Ctr., Bethesda, MD, Rept. No., SPD-S. Kaliski 670-01, 33 pp (Apr 1976)Dept. of Mechanics of Continuous Media, Institute AD-A027 377/1GAof Fundamental Techno Logical Research, PolishAcademy of Sciences, Swietokrzyska 21, 00)49 Key Words: Floating bodies, Damping coefficients, FiniteWarsaw, Poland, Bull. Acad. Polon. Sci., Ser. Sci. element technique, Oceans

Tech., 24 (6), pp 293-297 (1976) 5 refs Numerical results of the added man and damping coefficients

of vertical axisymmetric bodies on or under the free surfaceKey Words: Plates, Elasticity theory, Elgenvalue problems, are presented. The excitation forces on these bodies due toNatural frequencies an Incident regular wave systern are also computed, The nu-

merical scheme employs a localized finite-element method,The problem of natural vibrations is solved for a rectangular which is based on the theory of the calculus of variations.parallelepiped and a plate, on the basis of equations of the The excitation forces and moments on a submerged half.theory of elasticity, taking into consideration the linearized spheroid lying on the bottom are computed and comparedquantum effects which are Introduced by the methods of with the results obtained by others."classical" approach to quantum mechanics. The aigen-vales of the boJndery value problems are calculated.

77.280

T 7 Experiments on Transition to Turbulence in an17-a278 Oeillatory Pipe FlowWave Generation h4 n Elastic Half-Space by a Nor. M. Hino, M. Sawamoto and S. Takasumal Point Load Moving Unifornly Over the Free Dept. of Civil Engrg., Tokyo Inst. of Technology,Surface O-okayama, Meguro-ku, Tokyo 152, Japan, J. FluidA. Ungar Mech., 75 (2), pp 193-207 (May 27, 1976) 9 figs,Nati. Res. Inst. for Math. Sci., CSIR, Pretoria, South 15 refsAfrica, Intl. J. Engr. Sci., 14 (10), pp 935-945 (1976)16 refs Key Words: Fluid-filled containers, Pipes (tubes), Turbu.

lanceKey Words: Half space, Elastic properties, Moving loads,Wave propagation Experiments on transition to turbulence in a purely oscil-

latory pipe flow were performed. Three types of turbulentThe motion is determined of an elastic, homogeneous and flow regime were detected: weakly turbulent flow, condi-isotropic half-space, excited by a normal point load travel- tionally turbulent flow and fully turbulent flow. In theing uniformly over the free surface. Using the differential conditionally turbulent flow, turbulence is generrted sud-transform technique, exact, closed expressions in terms of denly In the decelerating phase and the profile of tne velocityalgebraic functions are found for the displacements at any distribution changes drastically. In the accelerating phase, thepoint of the half-space. flow recovers to laminar.

54

INELASTIC VISCOELASTIC(Also see No. 340)

77-281 77-283Dynamic Plastic Analysis of Ductile Fracture Vibration Analysis of Viscoelastic Bodies withthe Charpy Specimen Small Low TangentsD.J. Ayres Z, HashinCombustion Engineering, Inc., Windsor, CT 06095 School of Engineering, Tel-Aviv Univ., Israel, Rept.Intl. J. Fract., 12 (4), pp 567-578 (Aug 1976) 15 figs, No. TAU-SOEi340-76, Scientific-4, AFOSR-TR-76-12 refs 0830,22 pp (Apr 1976)

AD-A027 703/8GAKey Words: Dynamic plasticity, Fracture properties, Finiteelement technique, Computer programs Key Words: Viscoelastic media, Forced vibration, Composite

materialsA two-dimensional elastic-plastic dynamic finite elementanalysis of the precracked Charpy V-notch specimen is per. The correspondence principla for vibrations of viscoelasticformed using the MARC finite element computer program, bodies is specialized to the case of small loss tangents, resul-The results of the analysis are in good agreement with data ting in considerable simplification: analytical evaluation ofproduced from the instrumented precracked Charpy test. oscillatory fields is greatly simplified: peak frequencies andThis agreement verifies the computational techniques for peak amplitudes under forced vibrations can be simply andthis type of analysis and provides a much more detailed directly determined; numerical solution of viscoelasticview of the phenomena of ductile fracture. The analytical vibration problems becomes no more complicated thanprocedure developed will assist in the formulation and evalua- that of elastic problems. Similar simplifications result fortion of ductile fracture criteria and the application of these computation of real and imaginary parts of effective complexcriteria to pressure vessel and structural analysis. modull of composite materials.

EXPERIMENTATION77.282Application of an Energy Balance and an EnergyMethod to Dynamic Crack PropagationW. Doll BALANCINGInstitut fLJr Festkorpermechanik, 78 Freiburg i Br., (Also see Nos. 407,408, 409, 410,412, 413)Republic of Germany, Intl. J. Fract., 12 (4), pp 595-605 (Aug 1976) 8 figs, 27 refs

77-284Key Words: Dynamic plasticity, Fracture propert.- The Dynamics of a Hlooke's Joint Gyroscope with

Man UnbalanceThe speeds of fast running cracks in a range of quasi-brittle J.S. Burdess and L. Maundermaterials are measured as a function of the dynamic strain Univ of Newcastle upon Tyne, England, "Conf. onenergy release rate. From the results, the heat output asso- Vihrations in Rotating Machinery," The Institutionciated with the plastic work at the crack tip is calculated asa function of crack speed using an energy balance, and com- of Mechanical Engineers, Univ. of Cambridge, (Sept.pared with the heat outputs determined experimentally 15-17, 1976) op 195-199,2 refsusing an energy method. A generally good agreement is foundbetween the calculated and the experimentally measured Key Words: Gyroscopes, Unbalanced mass responsevariation of heat output with crack speed.

The dynamical characteristics of a Hooke's Joint Gyroscopewith mass unbalance in both the rotor and the gimbal aredescribed. For a constant rate input it is shown that the idealgyroscope may operate as a device for measuring eitherangular rates of turn or angular displacements dependingupon whether the gimbal suspension is tuned to the rotorspeed.

55

DIAGNOSTICS the VIBROSEIS system, is described. The monitoring trans-(See Nos. 304, 369,404, 405, 406, 445) ducers and the associated calibration system were designed

to operate in the 5- to 225-Hz frequency range and to mini-mize several problems associated with the particular source.The system was Installed, calibrated at sea, and performed

FACI LITI ES satisfactorily during the ocean exercise.

(Also see Nos. 400, 426)

77-285 77-287A New Acoustics Centre for Noise Investigations Resonant Pressure InstrumentationA Newi coutic Ce nr e fonies G.M. Schusterby Simulating Real Life Conditions Ph.D. Thesi;, Polytechnic Inst. of New York, 225 ppE. Schroder (96Autornobiltech. Z., 78 (7/8), pp 345-349 (July/ (1976)Aug 1976) 6 rigs, 15 refs(In German) Key Words: Resonators, Measuring Instruments, Design

techniques, CalibratingKey Words: Test facilities, Motor vehicle noise

This dlssertation Is concerned with the development ofDetailed noise investigations on a vehicle are carried out in analytical techniques which can be usefully applied to thethe new Acoustics Centre In Cologne, Merkenich. The ane- design and calibration of resonant pressure transducers,choic chamber, equipped with built-in two-axle chassis with emphasis on those Instruments which use an elasticdynamometer of latest design is described. The adjacent re- resonator as the basic senior. The dissertation shows howverberatlon room serves to determine the sound power and pressure-frequency relations may be developed for distri-to Improve the noise characteristics of vehicle components buted resonators, both by direct solution of linear differ-such as engine, transmission, and fan. A special opening ential equations and by approximation methods for nonlinearbetween the reverberation room and a transmitter room with cases. These relations are useful in establishing the operatingreverberation room characteristics allows specific sound- frequency range and pressure sensitivity as functions oftransmission loss investigations to be made on damping resonator dimensions and elastic properties, so that differentmaterials fitted to body parts such as vehicle dash panel resonator crinfigurations may be compared.assemblies.

77.288INSTRUMENTATION

(Also see No. 292) Analysis of Induced EMF in Vibrating-Sample Mag-netometersE.E. Brag,1 and M.S. Seehra

77286 Physics Dept., West Virginia Univ., Morgantown,An Acoustic Monitoring System for the Vibroseis WV 26506, J. Phys. E. (Sci. Instr.), 9 (3), pp 216-

An Aousic Mnitrin Sysem or he Vbroess 222 (Mar 1976) 13 figs, 10 ref sLow-Frequency Underwater Acoustic SourceA.M. Young and G.D. HugusNaval Research Lab., Washington, D.C., Rept. No. Key Words: Vibration measurement, Measuring instrumentsNRL.MR-3316, 22 pp (Junej.76) The theoretical induced EMF for vibrating-sample magnet-

AD-A028 239/2GA ometers is derived. Assuming a point magnetic dipole sample,

located in a general position with respect to a single-turn

Key Words: Underwater sound transmission, Acoustic mea- circular pickup coil, the instantaneous flux cutting the coil

surement, Measuring instruments and the induced EMF due to sample motion are found tobe linear combinations of elliptic integrals of the first and

One problem normally associated with the use of Low- second kind. An alternative representation of the flux by

frequency, high-power underwater acoustic sources is the a power series is also given. These results can be applied to

ability to iccurately monitor the generated sound pressure arbitrary versions of magnetometers by specifying coil

level. The design, fabrication and installation of an acoustic size and sample position and orientation. The effect of sam-

monitoring system for a large hydraulically actuated source, pIe misalignment on the induced EMF is computed. Acomparison of the four systems is also made.

56

------------

TECHNIQUES 77.291(Also see Nos. 308, 317, 318, 364, On the Use of Acoustical Holography to Locate

371, 387,395, 422) Sound Sources on Complex Structures

E.E. Watson and W.F. King, IllEnvironmental Acoustics Div., Universa' )il Products,

77-289 Wolverine Tube Div., Decatur, AL, J. oound Vib.,

Comparison of Methods for Measurement of Rever- 48 (2), pp 157-168 (Sept 1976) 14 figs, 15 refsberatious Time Sponsored by the Naval Sea Systems CommandT.E. Vigran and S. SorsdalDept. of Acoustics, The Norwegian Inst. of Tech-

Key Woids: Noise source identification, Testing techniques,nology, Trondheim, Norway, J. Sound Vib., 48 (1), Fans, Acoustic holographypp 1-13 (Sept 8, 1976) 10 figs, 9 refs

Until recently, it was extremely difficult to detect and locatefar-field sound sources on complex structures. An experimen-

Test facilities tel technique is presented in this paper for the identification

of such sound sources on complex vibrating structures.

Five methods for automatic measurement of reverberation Results are given for a preliminary investigation designed to

time are studied with respect to the mean value and the determine the feasibility of plying a modified form of

variance of it. The measurements were performed in two acoustical holography to the location of sound sources on

reverberation chambers. For each 1/3-octave band and micro- ventilator fans. Such sources are identifed on s thrce-bledod

phone position one hundred decay signals were evaluated. ventilator fan. Optical and computer sound soure recon-

It is felt that ,i automatic method should be recommended structions are compared and analyzed.

to supplement the "straight line" method outlined in existingstandards.

77.290 77-292

Shock.Front Loading Method for Studies in Dynamic Meamring Noise .. The State.of-the.Art

Photoelasticity A.J. Schneider

A.W. Miles B & K Instruments, Inc., Cleveland, OH, 44142,

Dept of Mech. Engrg, Univ. of Cape Town, Ronde- SAE Paper No. 760672, 8 p, 8 figs, 3 refs

bosch, 7700, Cape Province,, South Africa, Exptl.Mech., 16 (9), pp 349-355 (Sept 1976) 11 figs, Key Words: Noise measurement, Measurement techniques,17 refs -Measuring instruments

New instruments and new measurement concepts continually

Key Words: Photoelastic analysis evolve in support of programs to reduce operator noiseexposure and produce noise emission. The purpose of this

A versatile technique for applying a well-defined dynamic paper is to review new equipment and nw measurementload to models for studies in dynamic photoelasticity is practices which can assist in the direction of programs todescribed. The method utilizes the shock front produced in develop quieter vehicles. Opportunity is taken to also discussa gas-dynamic shock tube to apply a load to models by the closely associated field of whole-body vibration.direct normal impact. Tho principles and scope of the meth-od is suited to studies where simple variation and accuratedetermination of the load-cycle parameters, as well as precisereproducibility, are necessary. The method, in addition,permits close-field study of the initial response of materialsto dynamic loading to be undertaken.

57

COM PONENTS Key Words: Beams, Viscoelastic properties, Torsional vibra-COXI)M NENTStions

The complex dynamic shear modulus of soft polymericmaterials ray be determined in principle at low and audio

ASORBERS frequence- from torsion pendulum and torsional resonance(Also see No. 399) experiments on metal strips coated with the polymer. In

order to determine the polymer shear properties from such

experiments, it is necessary to know the torsional rigidity

77-293 of the twolayer compound beam. This is calculated in the

Structural Analysis and Design for Energy Absorption paper by using classical elasticity theory, for the particularcase when the metal shear modulus is much greater then

iImpact that of the soft coating. The theory of dynamic torsion testsE.H. Lee and R.L. Mallett is then briefly reviewed and experiments are suggested for

Dept. of Applied Mechanics, Stanford Univ.,, CA, determining the polymer dynamic shear properties. A dis-

Rept. No. SUDAM-75-15, DOT/TST-76/44, 215 pp cussion is also given of the effectiveness of polymer coatings

(Dec 1975) as a damping treatment for torsional vibrations.

PB-254 801/4GA

Key Words: Collision research (automotive), Beams, Shells, 77-295Energy absorption Dynamics of Timoahenko Beams Conveying Fluid

M.P. Paidoussis and BE. LaithierA general assessment of the nature of the dynamic problem Dept. of Mech. Engrg., McGill Univ., Montre!,of analyzing the collision of a vehicle Is first given. A vehicle U Mcommonly comprises an open structure of beams and thin Canada, J. Mech. Engr. Sci., 18 (4), pp 210-220

shells of ductile metal and the kinetic energy to be absorbed (Aug 1976) 8 figs, 13 refsas energy of deformation is usually sufficient to cause plasticflow. This Is generally localized on hinges and Is associated Key Words: Beams, Pipes (tubes), Timoshenko theory,with appreciable elastic deformation. Thus elastic-plastic Fluid-filled containers, Finite difference theory, Variationaltheory is needed including geometrical and material non- methodslinearities. Integration through the history of the motion isneeded for solution. Theorems are given which provide ap- The dynamics of pipes conveying fluid is described by meansproximations to the maximum plastic deformation and the of the Timoshenko beam theory. The equations of motionduration of plastic flow without determining the whole are derived and solved (a) by a finite-difference technique,solution. Application of these to simple models is proseited. and (b) by a variational method. It is shown that the latterThe theorems are also opplied to determine the stiffness of is the more efficient method. The eigenfrequencies of thea structural element so that it will transmit Impulse to the system and its stability characteristics are compared withrest of the structure without itself deforming appreciably, results obtained previously using the Euler-Bernoulli beama so-called stiff interface. The ability of porous metal to theory, and it is shown that in certain cases (e.g. short pipes)absorb energy of deformation is also analyzed by solving the the two sets of results diverge. Experiments indicate that themicroscopic problem of collapse of the cavities. The overall present theory is more successful in predicting the observedmacroscopic deformation laws can then be determined, behavior. Furthermore, the present theory shows that, inTests of the behavior of porous metal in Impact are presented some cases, cantilevered pipes may lose stability by buckling,and related to quasistatic measurements of its deformation whereas previous theories indicate that the system alwaysproperties. Conclusions from these studies and recommenda- loses stability by flutter.tions for new work are given.

BEAMS, STRINGS, RODS 77.2%(Also see Nos. 245, 272, 293, 313, 321. 341) Nonlinear Periodic Response of a Slender Beam-

ColumnB.P. Shafer

77.294 Ph D. Thesis, Univ of Denver, 183 pp (1976)

Dynamic Torsion of a Two-Layer Viscoelastic Beam UM 76-27, 873A F. Johnson and A. WoolfNatI. Phys Lab ,, Teddington, Middlesex TWI 1 OLW, Key Words: Beams-columns, Periodic response, NonlinearEngland, J Sound Vib., 48 (2), pp 251-263 (Sept response

(1976) 3 figs, 10 refs

58

The nonlinear dynamic response of a perfect beam-column 77-299subjected to loading by axial stress waves was studied. New- Three-Dimensional Stochastic Response of Offshoreton's method was used to solve the nonlinear equations itera- Towers to Wave Actiontively and to determine the lateral displacement and axialstrain as functions of time. An unstable linear approximation B. Berge and J. Penzienwas used to start the iterative procedure, and to guide the Structural Engrg. Lab., California Univ., Berkeley, CAsearch of the frequency domain for interesting nonlinear Rept. No. UCSESM-75-10, 158 pp (Oct 1975)phenomena. PB-254 n49/OGA

Key Words: Off-shore structures, Towers, Stochastic proces-

77-297 se

Corpasn of Solution Methods for Composite A theory is developed to calculate the dynamic response of

Material Beam Response Due to Impact off.shore towers to random wave forces. Vibrations areP.C. Chou and W.J. Flis considered simultaneously for translations in the orthogonal

Drexel Univ., Philadelphia, PA, Rept. No. NADC- tiorizontal directions and for rotations about a vertical axis.

76093-30, 59 pp (Dec 1975) The idealized structure for the dynamic analysis has its mos-Ases lumped at discrete horizontal levels. The ocean waves areconsidered to be a zero mean stationary ergodic Gaussian

random process described by the directional wave spectrum,Key Words: Beams, Composite materials, Impact response which specifies the distribution of wave energy with respect(mechanical), Computer programs to frequency and direction.

Several solution methods are compared as to their usefulnessIn treating impact problems. The problem used as a basisfor comparS, :n involves a simply supported beam Impacted 77.300at mid.span by a mass. The early.time and late-time structural Vibrations of Suspended Cablesresponses of the system have been computed by (1) an analy- J.G. Galetical solution, (2) the NASTRAN finite-element, structural.analysis computer program, using both the direct integrationmethod and the modal analysis method, and (3) the method UM 76-23,515of characteristics. The results have been compared. Thechief conclusions are that the method of characteristics Key Words: Cables, Equations of motionis the most accurate method for early-time. and that theNASTRAN modal method Is best suited for late-time struc. The equations of motion of an elastic cable suspended intural response. a viscous medium from two arbitrary points are derived.

These are then linearized for the undamped, inextensiblecase, and it is found that the equations describing the in-plane motion of the cable are then independent of the equa-

77-298 tion describing the out-of.plane motion of the cable. UsingEffects of Support Flexibility on the Stability of a the equations developed for the normal mode motion, aBeam Under a Follower Thrust and Inertia technique is presented for determination of the displace-JJ Wu ments and tensions throughout a cable when one and is

subjected to a prescribed tangential displacement of knownWatervliet Arsenal, Waterviet, NY, Rept No frequency. Maximum cable tensions are presented as aTR-76024, 20 pp (June 1976) function of the forcing frequency for a variety of cable

AD-A028 089/1GA geometries.

Key Words: Beams, Follower forces, Flexible foundations,Flutter, Stability analysis 77-301

The stability behavior of a slender structure, modeled by a An Experimental Investigation of Wave Propagationbeam, subjected to a follower force at one end and restrained in a Rubber String Impacted by a Projectileat the other, is studied in detail in this paper. Eigenvalue C. Riegel and J.L. Nowinskicurves which dictate the stability behavior are plotted for Univ of Delaware, Newark, DE 19711, Intl. J.several combinations of parameters. Nonlinear Mech., 11 (4), pp 229-237 (1976) 8

figs, 9 refs

59

4

Key Words: Strings, Elastomers, Dynamic tests Amer. Soc. Lubric. Engr., 19 (4), pp 267-272 (Oct

1976) 8 figs, 13 refsTransverse perpendicular impact on long rubber strings Spnsred by NASAwas studied using a special apparatus consisting of a sys-

ten of oscilloscopes, stroboscopes, camera and air pressuregun. The rubbers used in tests came from two manufacturers Key Words: Roller bearings, Elestohydrodynamic properties

and were classified as pure gum rubbers. Some of the resultsare compared with the theoretical data for the Mooney- The life and load carrying capacity of roller bearings are

Rivlin and the lsihara-Hashitsume-Tatibena-Zahorski mater- highly influenced by the sliding velocities at the contactsials, and others with the results of theoretical equations upon between rollers and races. In this investigation, a compu-inserting experimental data. tational model was developed to predict traction forces,

forces on the cage, sliding and spinning speeds of the rollerwhen the bearing is lubricated with a fluid of known trac-tion coefficient characteristics as functions of pressure and

77-302 velocities.Dynamic Mechanical Properties of Cotton and OtherFibresJ.L. Woo 77-304Ph.D. Thesis, Univ. of New South Wales, Australia, Information from Bearing Vibration(1976) R.C. Hemmings and J.D. Smith

R.H.P. Bearing Res. Centre, Chelmsford, England,"Conf. on Vibrations in Rotating Machinery," The

Key Words: Fibers Institution of Mech. Engrs., Univ. of Cambridge,

The dynamic elastic modulus of a polymer is a fundamental Sept. 15-17, 1976, pp 117-121,4 figs.

physical property which is specifically related to opticalbirefringence and molecular orientation. Such features Key Words: Ball bearings, Diagnostic techniques, Vibrationof natural fibres as their extremely limited length, lrregu, measurement, Tasting techniqueslarity of shape and area of the fibre cross-section, relativelylow yield point, spatially indeterminate geometry, fibre The standard test methods for bell bearings are described andfineness and flexibility, and the time-dependence of the fibre the relevance of testing to Installation noise problems ismechanical properties, constitute problems which must be discussed. It is shown that even the best current test specl-

overcome before any meaningful and reproducible method fications are inadequate for use in predicting vibration levels

can be developed for the determination of the dynamic in situ. Variation of test speed allows extraction of furthermechanical properties of these fibres, Following a critical information about bearing errors and resonances to assist

review of applicable physical principles and available experi- design. Further information about the bearing inner ring

mental techniques, the acoustic pulse propagation method can be extracted from the measured vibration by making use

was chosen as the most suitable principle for the present of sampling techniques: Conventional "whitewashing" and

application. In order to satisfy a set of imposed criteria, a wave analysis techniques are inadequate if used directly but

kinematic solution was obtained on which the design and if the signal is first sampled It may then be averaged to yield

construction of a constant-fibre-strain scanner was based. information about the Inner and outer groove shapes separ-

Associated small tools, experimental techniques and pro- ately. The technique is described and its use and limitationscedures have been devised. The dynamic modulus-strain are discussed.

curves for cotton, wool and jute fibres have been obtained.By using numerical integration, the dynamic stress-straincurves were constructed from the dynamic modulus-strain

curves, regarding the latter as the derivetive stress-strain 77-305curves for the specimen. A Theoretical and Experimental Investigation Illus-

trating the Influence of Non-Linearity and Miaalign.BEARINGS ment of the Eight Oil Film Force Coefficients

R.H Bannister

Cranfield Inst. of Technology, England, "Conf. on

77-303 Vibrations in Rotating Machinery," The Institutionsof Roller Bearings Considering Elato- of Mech. Engrs., Univ. of Cambridge, Sept. 15-17,

hydrodynamic Forces 1976, pp 271-278, 9 figs, 5 refs

Molna, D M Sanborn and W.0 Wier Key Words: Journal bearings, Hydrodynamic excitation,Univ of Carabobo, Valencia, Venezuela, Trans Lubrication

60

The results of a combined analytical and experimental ives. Key Words: Flutter, Turbine blades, Wind tunnel tests,tigstion, into the hydrodynamic oil film coefficients oper- Computer programsating in the laminar region are presented. An improved meth-od for analyzing a dynamically loaded journal is developed, A series of experimental measurements of airfoil pressuretaking into account the second order coeffiinets characteriz- distributions were made in an eleven-airfoil oscillating cas-ing non-linearity, whereby the theoretical elliptical whirl cade wind tunnel. The airfoils were oscillated in pitch aboutorbit can be deduced for any given periodic disturbing force. mid-cord pivots. A computer program was adapted to calcu.The investigation is also extended to take into account the late oscillating lift and moment coefficients on the assump-effects of miseligning couples applied in the veritcal plane, tions of incompressible flow, flat plate airfoils, fixed separ-illustrating that for this kind of misalignment the oil film ation point, zero pressure amplitude in the separated regioncoefficients are increased, thus having a stabilizing effect on and In the wake, and all streamlines parallel to the airfoilthe bearing, surfaces.

77.306 77.308Stabilization of Turbomachinery with Squeeze Film Experimental Methods for the Study of the VibratoryDampers - Theory and Applications Behaviour of Steam Turbine BladesE.J. Gunter, L.E.Barrett and P.E. Allaire R. BigretUniv. of Virginia, Charlottesville, VA, In: "Conf. on GETT, St. Rateau, 93120 La Courneuve, FranceVibrations in Rotating Machinery," The Inst. of "Conf. on Vibrations in Rotating Machinery," TheMech. Engrs., Univ. of Cambridge, Sept. 15-17, 1976, Inst. of Mech. Engrs., Univ. of Cambridge, Sept.14 figs, 13 refs. 15-17, 1976, pp 159-164, 5 figs.

Key Words: Turbomachinery, Periodic response, Transient Key Words: Turbine blades, Testing techniques, Measure-response, Squeeze film bearings, Rotor-bearing systems ment techniques, Vibration response

This study Investigates the steady-state and transient response The life of blading is greater, the smaller the tensile stressesof the squeeze film damper bearing, Both the steady-state which result from aerodynamic disturbances. During theand transient equations for the hydrodynamic bearing forces running of turbines where the local strains in the blles areare derived. The steady-state equations are used to determine measured, power spectral densities can be determined andthe damper equivalent stiffness and damping coefficients, stresses can be calculated. Since at the design stage, theThe coefficients are used to find the damper configuration evaluation of the stresses is still marked by some uncertaintywhich will provide the optimum support characteristics due to the difficulty of knowing precisely the exciting forcesbased on a stability analysis of the rotor-bearing system. and the exact shape of the natural modes, this paper deals

The effects of end seals and cavitated fluid film are included, with frequency criteria. A theoretical experiment procedureThe transient analysis of rotor-bearing systems is performed which enables one to know the natural frequencies of theby coupling the damper and rotor equations and integrating discs obtained Is discussed.forward in time. The effects of unbalance, bearing cavitation

and retainer springs, aerodynamic forces, and internal fric-tion are included in the analysis.

77-309The Characteristics, Prediction and Test Analysis of

BLADES Supersonic Flutter in Turbofan EnginesD.G. Halliwell

Rolls-Royce (1971) Limited, Derby Engine Div,

77-307 England, "Conf. on Vibrations in Rotating Machine-ry," The Inst. of Mech. Engrs., Univ. of Cambridge,

Stalled Flutter Sept. 15-17, 1976, pp 181-185, 6 figs, 3 refs

R A. Arnoldi, F.O. Carta, R H. Ni, and A.St. HilairePratt and Whitney Aircraft, East Hartford, CT, Rept Key Words: Turbine blades, Flutter, FansNo PWA-5420, AFOSR-TR-76-0829, 73 pp (May1976)AD-A027 869/7GA

61

The advent of high tip Mach Number, high work blading in This paper presents an analysis for the determination of thethe front fan stages of modern aero gas turbine engines has lift fluctuation on flat-plate-airfoils in cascade due to trans.led to the recognition of a self excited flutter phenomenon verse and chordwise gusts. The acceleration potential methodin the high speed unstalled regime of the fan operating is used in combination with conformal mapping method.characteristics. This dangerous stress condition, unlike a This method gives an utterly simple analysis even for cas-resonance, cannot be avoided by passing through the af- cades, without the complicated integral equations inevitablefacted speed range. This work, during design and develop- for the determination of vortex distribution in velocity-ment, has been devoted to obtaining a better understanding field analysis. The computational results show that the pitchof the aero-elastic mechanisms involved. The subject is intro- and the stagger angle of the cascade have much influenceduced by describing the vibration modes of unstalled, super- upon the magnitude of the fluctuating lift, but not so muchsonic flutter in fan assemblies having part-span shrouds or upon phase angle.clappers. The vibration pattern is seen to consist of discreteradial and circumferential nodes in the blade-disc assemblywhich can rotate relative to the rotor. A characteristic ofthis type of flutter is the selection of a unique, least-stable 77-312mode for any particular fan. Effect of Phaae Angle on Multibladed Rotor Flutter

V.R. Murthy and G.A. PierceDept. of Mech. Engrg. and Mechanics, Old Dominion

77-310 Univ., Norfolk, VA 23508, J. Sound Vib., 48 (2),Studies to Gain Insight into the Complexities of pp 221-234 (Sept 1976) 5 figs, 14 refsBlade Vibration PhenomenaD.J. Ewins Key Words: Rotary wings, Flutter, HelicoptersImperial College of Science and Technology, London,,Ime ld, Conege on Vciban in oT chlogyLo on- A theoretical technique for predicting the flutter charac-teristics of a helicopter rotor is presented. The effect of phasery," The Inst. of Mech. Engrs., Univ. of Cambridge, angle on flutter speed of a two-bladed rotor in hovering andSept 15-17, 1976, pp 165-172, 6 figs, 7 refs axial flight is determined. For this purpose, a uniform and

untwisted rotor blade with coupled flapwise bending and

Key Words: Turbine blades, Turbomachinery, Natural torsional degrees of freedom is considered. The transmissionfrequencies, Mode shapes, Test data matrix method is used to obtain the natural vibration char-

acteristics of the system. An unsteady aerodynamic theory is

This paper summarizes a series of investigations which have used to obtain the aerodynamic loading in compressible

been made in order to understand, explain, and predict the flow.

types of vibration response characteristics which are exhi-bited by turbomachine blades in their operating environment.The studies have been made using simplified but closelyrepresentative models of complete blaced disc assemblie., COLUMNSboth with and without shrouding. Using these models, it (Also see No 296)is possible to demonstrate the vibration characteristics ofthis type of assembly, both in terms of the natural frequen-cies and mode shapes and also in terms of the vibration 77-313response under typical excitation conditions. Influence of an Elastic End Support on the Vibration

and Stability of Beck's ColumnC Sundararajan

77.311 Foster Wheeler Energy Corp., Livingston, NY, Intl.The Unsteady Forces on Flat-Plate-Airfoils in Cas- J. Mech Sci., 18 (5), pp 239-241 (May 1976) 4 figs,cade Moving through Sinusoidal Gusts 8 refsS Murata and Y. TsujimotoDept of Mech. Engrg , Osaka Univ , Yamada-Kami, Key Words: Columns, Cantilevers, Elastic foundations,Suita, Osaka, Japan, Z Angew Math. Mech., 56 (5), Vibration responsepp 205-216 (May 1976) 16 figs, 12 rs The influence of an elastic support on the vibration and

stability of a non-conservatively loaded (follower force)Key Words: Rotor blades (turbomachinery), Flutter undamped, linearly elastic system is studied. It is found that

the support may destabilize the system in certain cases.

62

77-314 LINKAGES

Wind Induced Vibrations of Self-Supporting ConicalColumns with End ManL.J. Scerbo 77-316Ph.D. Thesis, Polytechnic Institute of New York, Vibrational Loading of Mechanically Fastened Wood262 pp (1976) JointsLIM 76-23,443 T.L. Wilkinson

Forest Products Lab., Madison, WI, Rept. No. FSRP.Key Words: Columns, Wind-induced excitation, Mathematical FPL-274, 15 pp (1976)models AD-A026 642/9GA

This study presents a mathematical model for predictingthe wind induced vibrations of self-supporting, linearly ta- Key Words: Joints (junctions), Vibration responsepered, truncated, conical columns with end mass. The col-umns considered can be made up of one or more linearly The paper presents the results of vibrational loading oftapered uniform wall thickness sections, joined together nailed and bolted wood joints subjected to a range of fro-to form a continuum. Their dynamic behavior is simulated quencies and dead loads. Each joint was tested as a spring-by means of the Euler-Bernoulli bending differential aqua- mess system subjected to base excitation. Results indicatetion, which is applicable for small deflections of variable a considerable increase in joint stiffness under vibrationalcross section beams. loading as compared to static loading with very small a-

mounts of additional joint deformation.

CYLINDERS MECHANICAL(See No. 329)

77-317Technical Diagnostics of Driving Gear Elements

GEARS J. Mtller and D. TroppensWilhelm-Pieck-Universit~t Rostock, Dem. Rep. Ger-many, Maschinenbantechnik, 25 (8), pp 350-353

77.315 (Aug 1976) 4 figs, 16 refsStatistical Analysis of Dynamic Loads on Spur Gear (In German)Teeth (Effect of Shaft Stiffness)T Tobe, K. Sato, and N Takatsu Key Words: Diagnostic techniques, CrankshaftsDept of Engrg , Tohoku Univ , Japan, Bull, JSME, After describing technical diagnostics, which has developed19 (133), pp 808-813 (July 1976) 13 figs, 4 refs into a special branch of maintenance, the authors discuss

its application in the analysis of driving gear elements andKey Words: Gears, Shafts, Statistical analysis, Dynamic compile all known diagnostic techniques for the crank driveresponse elements of high speed diesel motors. Only those parameters

are investigated which are necessary for the determinationIn the previous paper, a statistical method of evaluation of of damage.the relation between transmission errors and dynamic loadson a pair of spur gears was established neglecting shaft stiff-nesses. A more advanced study considering the shaft stiffnessis developed in this paper. A Fokker-Planck equation is 77-318derived and the mean and the variance of dynamic loads Driveline Vibrationsare calculated numerically by applying statistical lineari- J.B. Largezation to nonlinear terms. Monte-Carlo-Simulation is also General Motors Sthoot of Product Service, Generalperformed to find maximum dynamic loads by means of ananalogue computer. Motors Corp.,, Warren, MI , Fleet Maintenance and

Specifying, p) 46, 48-51 (Nov 1976)

63

Key Words: Motor vehicles, Ground vehicles, Trucks, Vibra- scheme for timewise integration. The resulting nonlineartion control, Diagnostic tecniques algebraic equations are solved by Newton's iterative method

to yield the deflection coefficients at all time intervals.Techniques for identifying and correcting truck driveline The present investigation essentially deals with the influencevibrations are described, of dynamic loads on the snap-through behavior of sandwich

panels. The effect of the initial curvature and core modulusis studied in detail. Static snap-through collapse loads are alsocalculated and compared with dynamic loads. As a check for

77-319 the present analysis the results obtained for the static caseThe Dynamics of a Two Rotor System with Slipping for a homogeneous plate are compared with the available

Clutch results.

J. StewartTwiflex Couplings, Ltd., England, "Conf. on Vibra- PIPES AND TUBEStions in Rotating Machinery," The Inst. of Mech. 77-321Engrs., Univ of Cambridge, Sept. 15-17, 1976, Escape Piping Vibrations while Designingpp 99-104, 7 figs, 3 refs J.C. Wachel and C.L. Bates

Southwest Res. Inst., San Antonio, TX, HydrocarbonKey Words: Power transmission systems, Couplings, Clutches, Processing, 55 (10), pp 152-156 (Oct 1976) 5 figs,Vibration control, Numerical analysis 8 refs

rhe motion of a two mass linear or non-linear system withsinusoidal torsional excitation is studied with special refer- Key Words: Piping systems, Beams, Vibration responseence to the presence of a slipping element, such as a centri-fugal clutch coupling, during both transient and resonant In this paper the non-ideal conditions which exist in pipingconditions. Numerical solutions using parameters typical of systems are compared to the ideal beam theory, in ordera medium power diesel transmission show the strong de- to predict accurately the mechanical response of piping sys-tuning action and consequently high damping efficiency of tems.the clutch in relation to a linear system. The model correctlypredicts the anticipated characteristics of the non-linearsystem, and is shown to be accurate for known solutions. 77322

A Composite Transfer Matrix.Synthesis Approachto Piping Dynamic Analysis

MEMBRANES, FILMS, AND WEBS P. Belier(See Nos 240, 245) Ph.D. Thesis, Polytechnic Inst of New York, 246 pp

(1976)UM 76-23,394

PANELSKey Words: Piping systems, Natural frequenciet, Modeshapes, Transfer matrix method, Computer programs

77.320Nonlinear Dynamic Buling of Sandwich Panels A composite transfer matrix-synthesis approach to pipingsystem dynamic analysis is described. The transfer matrixN.K. Adi Murthy and R.S. Alwar method is used to determine the dynamic characteristics ofDept of Appl. Mechanics, Indian Inst. of Tech- simple configurations considered to represent typical pipingnoloqy, Madras, India, J. Appl. Mech , Trans ASME, components. The developed data, tabular listings of transfer43 (3), pp 459-463 (Sept 1976) 4 figs, 13 refs matrix state vectors, include sufficient information to delin-

eate the first five natural frequencies and their associateddeflection and force mode shapes for five fixed-fixed com-

Key Words:, Sandwich panels, Dynamic buckling, Snap ponent configurations over a range of their span lengththrough problems parameters. These data will define the dynamical charac-

teristics of any three span configuration likely to occur inNonlinear dynamic buckling of simply supported rectangular piping runs. Used with the fixed constraint modal synthesissandwich panels with isotropic cores and wtn initial curva- method, they permit the ready analysis of the most gener,ture under transverse loads is analyzed. The large deflection alized piping systems. Specific examples of applications tosandwich panel equations proposed by Reissner are solved typical piping configurations, both simple and branched,using a trigonometric series spacewise and by Houbolt with an assessment of technique accuracy are included.

64

77-323 Key Words: Shells, Composite materials, Fiber composites,

Optimizing Valve Jet Size and Spacing Reduces Valve Missiles, Computer programs

Noise It has been postulated that anisotropic structures such as

C. Reed the fiber composites should be governed by kinematic

Masoneilan International, Inc., Norwood, MA., hardening. The feasibility of incorporating the changes of

Control Engineering, 23 (9), pp 63-64 (Sept 1976) degrees of anisotropy during the post-yielding deformations

3 refs of fiber composites is reported. The present report describes

the development of kinematic plastic function theory. The

method requires a considerable amount of computing time,Key Words: Valves, Noise reduction but upon careful reprogrmming, it is believed to be a power-

ful tool in the analysis of missile structures.

The concept of using a spaced-array of small fluid jets to

limit noise generation in process control valves has been

applied by the author to several sizes of actual hardware.The resulting design was found to be very effective in limiting 77-326

noise generation while using basic parts of the standard Vibrations of Thin Elastic Shell. A New Approachcage-guided valve. M.J.A. O'Callaghan, W.A. Nash and P.M. Quinlan

Dept. of Civil Engrg., Massachusetts Univ., Amherst,MA., Rept. No. AFOSR-TR-76-0731, 111 pp (Aug

77-324 1975) (AD-779 782)Pipewall Vibration Reveal ... Valve.Generated Noise? AD-A027 706/1 GAA.C. FagerlundFisher Controls Co., Marshalltown, IA., Hydrocarbon Key Words: Spherical shells, Natural frequencies, ModeProcessing, 55 (10),, pp 147-149 (Oct 1976) 7 figs shapes, Computer programs

Key Words: Piping systems, Valves, Noise generation The present Investigation extends the determination of

natural frequencies and mode shapes of free vibration of

Laminar fluid flow under steady pressure through a pipe of thin elastic plates to the case of a shallow elastic spherical

uniform cross-section can be almost noiseless. When flow is shell of n-sided polygonal plan-form. Natural frequencies

regulated, however, the valving may introduce varying and associated mode shapes together with boundary residuals

amounts of turbulence or even cavitation. These forms of (indicating how well the approximate solution has satisfied

energy propagating through the fluid as.-' fluctuating pressure boundary conditions) are readily displayed by the computer

force the pipewalls to vibrate. The vibrations in turn cause program offered. Results obtained in the present work are

pressure disturbances outside the pipe that radiate as sound. shown to be in excellent agreement with existing values for

When measuring sound generated by a single control valve, specific boundary conditions treated. The present approach

extraneous noise sources and reflected sound make interpre- involves only modest computer efforts but offers the signifi-

tation of the measurements difficult. This paper is concerned cant feature of rapid determination of how precisely the

with the conversion of pipe vibrations into an equivalent boundary conditions have been satisfied along each edge of

sound pressure level in ambient air. the shell.

PLATES AND SHELLS 77.327(Also see Nos 245, 277, 293, 341, 363) Free Vibration of Fluid-Coupled Coaxial Cylindrical

Shells of Different LengthsM.K. Au-Yang

77-325 Nuclear Power Generation Div., Babcock & Wilcox,Static and Dynamic Analysis of Thermoviscoelasto- Lynchburg, VA., J. Appl. Mech.,, Trans. ASME,

plastic Fiber-Reinforced Composite Shells in Missile 43 (3), pp 480-484 (Sept 1976) 3 figs, 9 refs

StructuresT.J. Chunq and R L Eidson Key Words: Cylindrical shells, Submerged structures, Fluid-

Dept of Mech. Engrg ,, Alabama Univ in Huntsville, induced vibration

AL, Rept. No. UAH-RR-182, 185 pp (May 1976)AD-A027 154/4GA

65

This paper considers the free vibration of two coaxial cylin. R.K. Dubeyders with finite lengths immersed in a restricted fluid me- Ph.D. Thesis, Polytechnic Inst. of New York, 84 ppdium. The natural frequencies in the fluid can be computed (1976)by including the virtual mass to its physical mass, Experimen-

UM 76-23,402tal results have been used to verify the analysis. Agreementbetween experimental and theoretical results is excellent.

Key Words: Shells, Periodic response

The steady state response of shell structures under axial and77.328 lateral excitation is reported. The two point boundary valueFree Vibrations of Circular Cylindrical Shells problem is solved by reducing the equations of motion to aV. Ramamurti and J. Pattabiraman system of first order differential equations, and applying theDept. of Appl. Mechanics, Indian Inst. of Tech, method of stepwise integration. The axial vibration trans-Mdp of60pp0.0 Mec, Indiau n Ist. of T(1), pmission characteristics for cylindrical shells as a function ofMadras 600036, india, J Sound Vib, ,48 (1), pp 137- circumferential mode number has been determined. The155 (Sept 8, 1976) 14 figs, 27 refs results for short and long cylindrical shells with various

boundary conditions have been presented as elements of aKey Words: Cylindrical shells, Free vibration, Finite element transfer matrix. The response of laterally excited cylindricaltechnique shells and containment vessels has been obtained and the

stresses, moments and disolacements along the shell generatorThe finite element method is used to predict the dynamic have been determined as a function of frequency.behavior of circular cylindrical shells in free vibrations. Asuitable shape function for the circumferential displacementdistribution has been proposed. This reduces the three-dimensional character of the problem to a two-dimensionalone. The simultaneous iteration method to determine the Dynamic Behavior of Fluid.Tank Systemseigenfrequencies and eigenvectors is utilized for solving the J.Y. Yangeigenvalue problem. The accuracy of the method has been Ph.D. Thesis, Rice Univ., 214 pp (1976)checked by verifying the results of known cases. Finally UM 76-21,727an experimental shell structure containing elastic ringswelded at the ends has also been analyzed and the experi-mental results compared with the theoretical ones. Key Words: Fluid-filled containers, Storage tanks, Earth-

quakes, Seismic exictation

Presented in this report are the results of a study of the77.329 hydrodynamic forces induced in fluid-filled storage tanksVibration of Orthogonally Stiffened Waffle Cylinders subjected to earthquake excitation and of the free vibrationSubjected to Initial Forces of the coupled fluid-tank systems. Throughout this study,,the fluid is assumed to be incompressible and inviscid. TheM.I. Baig and T.Y. Yang report can be divided into four parts: The first part dealsPurdue Univ., West Lafayette, IN., J Spacecraft and with the hydrodynamic forces induced In a rigid tank sub-Rockets, 13 (10), pp 618-623 (Oct 1976) 9 figs, jected to a lateral earthquake excitation. In the second part6 refs of the report an approximate analysis is presented for the

hydrodynamic forced induced in flexible fluid-filled tanks.The third part of the report deals with the free vibrational

Key Words: Cylindrical shells, Equations of motion characteristics of flexible circular cylindrical tanks. Thefourth and final part of the report deals with the axisym.

The kinetic energy expression is formulated for the ortho- metric free vibration of fluid-tank systems.gonally stiffened waffle cylindrical shell. The equations ofmotion are formulated by using Hamilton's principle forsimply supported edge conditions. The effect of initialmiddle-surface stresses is included. A variety of examples are 77-332performed, and results are compared, whenever possible, with Shells and Plates Loaded by Dynamic Moments withalternative solutions.

Special Attention to Rotating Point MomentsW. SoodelR.W. Herrick Laboratories, Purdue Univ.,, West

77-330 LafayEtte, IN 47907, J. Sound Vib., 48 (2), pp 179-Steady State Response of Shell Structures 188 (Sept 1976) 3 figs, 6 refs

66

Key Words: Shells, Plates, Rotors, Dynamic response These variational principles have a form which can be charac-terized as a nonlinear Rayleigh principle, finite-element

The response of thin shells to line or point moment excite- models are introduced and are used to implement the post-tion is formulated by way of distributed moment fields. buckling variational principle. Incremental amplitude solu-Twisting moments in the tangent plane are part of this tion schemes are developed to solve the corresponding finite-formulation. The approach is illustrated by using Love's element equations. Sample results are presented for thethin shell theory, but is valid for any other thin shell theory postbuckling beh.:,, cf circular plates.as well. Dirac delta functions are used to describe line andpoint moments. As a first example, the response of a plate toa rotating moment is evaluated and shown to be identicalto the solution obtained by Bolleter and Soedel by a Green 77-335function approach. The three-directional response of a cir- On the Non-Linear Dynamic Stability Problem forcular cylindrical shell to a rotating moment Is given as the Thin Elastic Platessecond example. S.D. Kisliakov

Inst. of Civ. Engrg. and Arch., Sofia, Bulgaria, Intl. J.Nonlinear Mech., 11 (4), pp 219-228 (1976) 7 figs,

77-333 7 refsFree Vibration of Two Span Rectangular Plates withSome Non-Claaaical Boundary Conditions Key Words: Dynamic stability, Plates, Parametric excitation

S. Mirza and B. SetiawanDept. of Mech. Engrg., Univ. of Ottawa, Ontario, In this paper the non-linear response of thin elastic plates

Canada, Intl. J. Mech. Sci., 18 (4), pp 165-170 under parametric excitation is investigated. A new analyticalmethod is proposed. It gives the possibility to obtain all the

(Apr 1976) 7 figs, 10 refs characteristic features of the phenomenon considered, which

are known from experiments - the existing of beats, their

Key Words: Rectangular plates, Free vibration, Spring-mass dependence on the excitation parameter, the Influence ofsystems the initial conditions, the typical character of the vibrations

In the different regions. Analog computer studies are carried

This paper deals with the vibration of two span rectangular out, and they show clearly the Influence of different pare-plates resting on linear and torsional springs. The two oppo- meters on the output of the problem considered.site continuous edges of the plate are considered as simplesupports. The resulting equations are general. Numericalresults have been obtained by varying spring stiffness and theaspect ratio. Some limiting cases of Interest have also been 77.336developed. Natural Frequencies of Orthotropic Rectangular

Plates with Varying ThicknesT. Sakata

77.334 Dept. of Mech. Engrg., Chubu Inst. of Technology,

- Finite Element Methods for Nonlinear Eigenvalue Kasugai, Nagoya-sub., Japan 487, J. Acoust. Soc.Problems and. the Poatbuckling Behavior of Elastic Amer., 60 (4), pp 844-847 (Oct 1976) 2 figs, 13 refsPlatesL C. Wellford, Jr and G.M. Dib Key Words: Rectangular plates, Variable cross section,

Civ. Engrg Dept., Univ. of Southern California, Los Natural frequencies

Angeles, CA 90007, Computers and Struc., 6 (4/5), An approximate formula is derived for the estimation of

pp 413-418 (Aug/Oct 1976) 6 figs, 16 refs the fundamental natural fruqency of the simply supportedorthotropic rectangular plate with thickness varying linearly

Key Wrds: Plates, Buckling, Eigenvalue problems, Finite in one direction. The accuracy of the formula and the

element technique influence of the flexural rigidity on the natural frequencyare discussed.

In the following paper new variational methods for the solu-

tion of problems of postbuckling of elastic plates are formu-lated, and approximate solutions for various cases are ob-tained using finite element methods. Initially, the post-buckling problem is phrased as a nonlinear eigenvalue prob- SPAN: A Computer Program for Static and Dynamiclem. Variational principles for the nonlinear eigenvalue Analysis of Stiffened Pates and Grillagesproblem of the postbuckling of structures are developed.

67

G.C. Mitchell, J.D. Clarke and C.S. Smith Key Words: Disks, Rotating structures, Flextural vibration

Naval Construction Research Establishment, Dun-fermline, Scotland, Rept. No. NCRE/R630, DRIC- This paper presents results of an investigation on the effect

of a transverse load on the stability of a spinning elasticBR-50857, 67 pp (Feb 1976) disk. The disk rotates at constant angular velocity and theAD-A026 865/6GA load consists of a mass distributed over a small area of the

disk, a spring, and a dashpot.Key Words: Computer programs, Plates, Grids (beam grids),Ship structural components, Finite element technique

A computer code for the static and dynamic analysis of flat RINGSplate grillages is described. The code was a finite elementdisplacement method incorporating a geometric stiffnessmatrix to calculste the response to static, cyclic or transient 77-340loads. Natural frequencies and buckling loads can also becalculated. A full input description is given including a An Analytical Formulation of the Forced Reqonessimplified data generation facility for regular gilllages. Input of Daped Ringsand output for some representative sample problems are Y.P. Luincluded. David W. Taylor Naval Ship Res. and Dev. Ctr.,

Annapolis, MD 21402, J. Sound Vib., 48 (1), pp 27-33 (Sept 8, 1976) 3 figs, 5 refs

77.338Free Vibration Analyais of Stiffened Plates Using Key Words: Rings, Composite structures, ViscoelasticFinite Difference Method damping, Forced vibration

G. Aksu and R. Ali An analysis In which an interaction formulation is used forDept. of Transport Technology, Univ. of Technology, the forced vibratory responses of a ring having a number of

Loughborough LE11 3TU, England, J. Sound Vib., mass segments adhered to it by a viscoelastic material Is48 (1), pp 15-25 (Sept 8, 1976) 9 figs, 8 refs presented. The mess segments are discretely distributed

around the circumference of the ring, and the excitation is aconcentrated vibratory radial force located on the surface

Key Words: Plates, Free vibration, Finite difference method of te ing. the m as segm ents a t e o be I feof the ring. The mass segents may not have to be Identical,nor do their distributions have to be uniform, The tnalysls

Free vibration characteristics of rectangular stiffened plates n di e etend to e om Te nsll

having a single stiffener have been examined by using the can readily be extended to more complicated nped shell

finite difference method. A variational technique has beenat a location midway between two mass segments for a given

used to minimize the total energy of the stiffened plate and ampedlsstem aeten as a e mets olution

the derivative appearing in the energy functional are replaced damped sstem are given as an example, These solutionsby finite difference equations. The energy functional Is compare very well with experimental data and theoreticalbymfinim t difference equations.pTheenergy functiol is results available. Also presented is a comparison of drivingminimized With respect to discretized displaement coin- ponmehiclmedcsfrtwda edytmsih

ponents and natural frequencies and mode shapes of the point mechanical impedances for two damped systems with

stiffened plate have been determined as the solutions of a di*rent thicknows of the viscoestic layers.

linear algebraic eigenvalue problem. The analysis takes intoconsideration inplane deformation of the plate and thestiffener and the effect of inplane inertia on the natural STRUCTURALfrequencies and mode shapeL The effect of the ratio ofstiffener depth to plate thickness on the natural frequenciesof the stiffened plate has also been examined.

77.341Structural Vibrations and Fourier Series

77-339 R. Greif and S.C. Mittendorf

The Stability of a Spinning Elastic Disk with a Dept. of Mech. Engrg., Tufts Unit,, Meuforu, MA

Transverse Load System 02155, J. Sound Vib., 48 (1), pp 113-622 (Sept 8,

W.D. Iwan and T.L. Moeller 1976) 5 figs, 14 refs

Div of Engrg and Appl Science, California Inst. ofTechnology, Pasadena, CA ,J. Appl. Mech., Trans. Key Wo,ds: Structur.l elemwnts, Beans, Plotes, Shells,TSMechnology, p CA ( 1 8 fich, Trns Variable material properties, Variable cross section, Con-ASME, 43 (3), pp 485-490 (Sept 1976) 8 figs, 10 refs pon,:nt mode analysis

68

A method is introduced for vibration analysis of a wide class Key Words: Shock absorbersof beam, plate and shell problems including the effects ofvariable geometry and material properties. The method is The performance of an impact absorber with a linear springbased on the discrete technique of component mode analysis, and a quadratic law damper is analyzed. The equation ofFor each of these components the mode shapes are written motion is non-dimensionalized and a closed form solution isIn terms of Rayleigh-Ritz expansions involving simple Fourier obtained. Chosen response variables are presented in terms ofsine or cosine serieL Due to the nature of these series, special a dimensionless parameter, the quadratic damping ratio. Theattention must be given to end point behavior In the model performance Is compared with that of an absorber withexpansions and in the derivatives of these modal expansions, viscous damper.This is done via the mechanisms of Stokes' transformation.Continuity between components is enforced with Lagrangemultipliers. The resulting frequency equation is exact and theassociated elgenvector contains a combination of force and ACOUSTIC ISOLATIONdisplacement types terms, Numerical solutions are found (See No. 260)by truncating the series and monitoring the frequencydeterminant on a computer.

NOISE REDUCTION(Also see Nos. 250, 323,353,354, 367, 368, 372,389, 402)

SYSTEMS77.344Muffling Hydraulic SystemsS.J. Skaistis and R.J. Becker

ABSORBER Systems Science Labs., Sperry Vickers Div., Sperry(Also see Nos, 379, 380, 381) Rand Corp., Troy, MI, Mach. Des., 48 (24), pp 124-

128 (Oct 21, 1976)

77.342 Key Words: Hydraulic equipment, Noise reduction

Dynamic Analysis of Impact Attenuation SystemsItiizing Plastic Deformationa (Case in which Effect Techniques for selecting, mounting and shielding hydraulic

of Strain.Rate Sensitivity is Cot.dered) components to avoid excess noise are described.

K. OhmataDept. of Engrg., Meiji Univ., Kawasaki, Japan, Bull.JSME, 19 (134), pp 884-901 (Aug 1976) 13 figs, 77.3456 refs Nonlinear Distortion in the Propagation of Intense

Acoudie NoiseKey Words: Shock absorption, Plastic deformation F.M. Pestorius and D.T. Blackatack

Appl. Res. Lab., Texas Univ. at Austin. Austin, TX,An impact attenuation system whose idealized static load- Rept. No. AFOSR-TR-76-0925, 15 pp (Mar 1973)displacement curve consists of an elastic range and unre- Sponsored by ONRstricted plastic flow is represented by a dynamically equiva-lent model which takes into account elasticity, viscosity, AD-A027 650/1GAplasticity and strain-rate sengitivity of the structure, and itsdynamic response to the impact by a moving mass is analyzed. Key Words: Sound transmission, Noise reduction

As sound pressure increases to 'finite' levels, nonlineareffects usually Ignored in acoustics problems become in-

77-343 crsesingly important. In this paper, plane waves propagating

Impact Absorber with Lin~ear Spring and Quadratic through a pipe are considered. In the first section the distor.Law Damper tion of a wave that is sinusoldal at the source is treated. InM.S. Hundal the second section the source signal is a pulse band-limited(500 to 3500 Hz) random noise. In both cas theoreticalDept af Mech Engrg., Univ. af Vermont, Burlington, results are compared with experimental measurementsVT, J. Sound Vib., 48 (2), pp 189-193 (Sept 1976) Extension of the analysis to the more practical case of4 figs, 6 refs outdoor propagation is indicated.

69

77-346 Key Words: Suspension systems (vehicles), Active isolation,

Meeting European Noise Regulations for Rubber- Cofputer programs

Tired Front End Loaders An active, computerized, adaptive control system is discussedD.F. Rudny for the control of vehicle vibrations by means of adjustmentPay line Div., International Harvester, SAE Paper No. of suspension parameters during the ride. The problem is

760599, 12 pp, 18 figs, 2 refs subdivided into two basic parts. One, optimization of vehiclesuspension parameters if forward speed, terrain statistics andvehicle descriptors are known a-priori. Two, methods ofevaluating terrain data received from sensors on board thevehicle and subsequently processing it for comparison withMajor legislation has been established in Europe governing the stored data contained in the memory.

the noise levels of heavy construction equipment. Level

limits, test procedures, date acquisition techniques, andfuture legislation of several European countries are presentedspecifically for rubber tired front end loaders. Techniques AIRCRAFTutilized In determining problem source components and (Also see Nos. 263, 436)methods employed in attenuation of the vehicle to complywith the existing legislation are also discussed. Special atten.tion Is given to the spectator level where the regulation limitsare most stringent. However, many of the modifications 77.349shown are also effective In reducing operator station noise Recursive Identification and Tracking of Parameterslevels, for Linear and Non-Linear Multivariable Systems

M. SidarAmes Res. Ctr., NASA, Moffett Field, CA 94035,

77-347 Intl. J.. Control, 24 (3), pp 361-378 (Sept 1976)Noise Reduction in the Generation of Energy from 6 figs, 26 refsBrown Coal as an Environmental Production Prob-lem Key Words: Parameter Identification, Aircraft, StabilityR. Junghans and P. Koltzsch analysisKDT, Bergakademie Freiberg, Germany, Technik(Berlin), 31 (8), pp 532-536 (Aug 1976) 7 tigs, The problem of Identifying constant and variable parameters

10 refs in multi-input, multi-output, linear and non-linear systems Isconsidered, using the maximum likelihood approach. An

(In German) iterative algorithm, leading to recursive identification andtracking of the unknown parameters and the noise covariance

Key Words: Noise reduction, Industrial facilities matrix, is developed. Agile tracking and accurate and un-bilaed identified parameters are obtained. Necessary condi-

After a short review of noise control regulations in the tions for a globally, asymptotically stable identificationDemocratic Republic of Germany, the application of model- process are provided; the conditions proved to be useful anding technology for noise control in the brown coal industry efficient. Among different cases studied, the stability derive-is described. Two examples are illustrated: one is a mathe- tives of an aircraft were Identified and some of the results arematical model for the investigation of noise propagation inpower stations, the other a model for the investigation of theacoustic effect of noise isolators in ventilation coolingtowers.

77-350ACTIVE ISOLATION Dynamic Behavior of an Aircraft Encountering

Aircraft Wake TurbulenceR.C. Nelson

77-348 Univ. of Notre Dame, Notre Dame, IN, J. Aircraft,Development of Computerized Active Vehicle Sus- 13 (9), pp 704-708 (Sept 1976) 10 figs, 13 refspensionsH K. Sachs and R.W. Siorek Key Words: Aircraft, Aerodynamic response, Vortex in-Dept. of Mech Engrg. Sciences, Wayne State Univ., duced vibrationDetroit, MI., Rept No. TACOM-TR-12126, 96 pp(Jan 1976)AD-A028 390/3GA

70

This paper deals with the dynamic behavior of an airplane rotors, helicopter rotors, lift fans, gearboxes, and drive

encountering aircraft wake turbulence. A digital computer engines. In this report, noise sources for each of these propul-

simulation was developed to study the response of an aircraft sors, gearboxes, and drive engines are identified and rank

flying into a trailing vortex wake. The simulation includes the ordered. The noise generating mechanisms for each of the

complete six degree-ol-freedom equations of motion, a des- propulsor noise sources identified are defined and systemati-

cription of the vortex velocity field, unsteady aerodynamics, cally catalogued. Three approaches to reduction of propulsor

and pilot control input. The parameters, varied in this simula- noise are discussed: changes in physical geometry, changes

tion, include the penetration angle, separation distance, air- in design operating conditions, and the use of acoustic

craft size (for both the penetrating and generating aircraft), treatments. Computerized and graphical procedures based on

and pilot control input (single- or multi-axes). Predicted methodology from the open literature and at United Tech-

vortex induced motions are presented for several probe nologies Corp., are presented for predicting aerodynamic

aircraft. performance of and noise from the V/STOL propulsorsidentif led In thi: study.

77.351On Sound Transmission into a Thin Cylindrical Shellunder "vight Conditions" V/STOL Rotary Propulsion Systems Noise PredictionL.R.and Reduction. Volume . Graphical PredictionDept. of Mech. and Aero. Engrg., Univ. of Missouri- aedhRduRoll. Rolla, MO 65401,, J. Sound Vib., 48 (2), Methodspp 265-275 (Sept 1976) 4 figs, 16 refs B aloiHamilton Standard Div., United Technologies Corp.,

Windsor Locks, CT., Rept. No, FAA-RD-76-49.2,Key Words: Cylindrical shells, Aircraft, Sound transmission, 299 pp (May 1976) No.. 389)

Mathematical models 299 pp (May 1976) (AD-A027 389)AD-A027 390/4GA

In the context of the transm!ssion of airborne noise into anaircraft fuselage, a mathematical model for sound transmis- Key Words: Aircraft noise, Engine noise, Vertical takeoffsion into a thin cylindrical shell is used to study sound aircraft, Noise prediction, Noise reduction, Graphic methodstransmission under "flight conditions":, i.e., under conditionsof external air flow past a pressurized cylinder at flight Graphical procedures for estimating noise and performancealtitude. Numerical results for different incidence angles are of free-air propellers, variable pitch fans with Inlet guidepresented for a typical narrow-bodied jet in cruising flight vanes, variable pitch fans with outlet guidc vanes, fixed pitchat 10 660m (35 000 ft) with interior pressure at 2440 m fans, helicopter rotors, tilt rotors, and lift fans are presented.(8000 ft). A comparison is made between no-flow sound Noise prediction methods for drive engines, gearboxes, jetstransmission at standard conditions on the ground to sound with and without bypass flow, as well as noise reduction andtransmission under flight conditions. performance losses for partly sonic inlets and duct linings

are also presented.

77-352V/STOL Rotary Propulsion Systems Noise Predictionand Reduction. Volume I. Identification of Sources, 77-354Noise Generating Mechanisms, Noise Reduction V/STOL Rotary Propulsion Systems - Noise Predic-Mechanisms, and Prediction Methodology tin and Reduction. Volume Ill. Computer ProgramB Maghoizi User's ManualHamilton Standard Div., United Technologies Corp B Maqlio.iWindsor Locks, CT., Rept. No. FAA-RD-76-49 1, Hamilton Standard Div , United Technologies Corp.,145 pp (May 1976) (AD-A027 390) Windsor Locks, CT., Rept. NO, FAA-RD-76-49-AD-A027 389/6GA Vol-3, 300 pp (May 1976) (AD-A027 389)

AD-A027 363/1 GAKey Words:. Vertical takeoff aircraft, Aircraft noise, Noisesource identification, Noise prediction, Noise generation Key Words: Aircraft noise, Engine noise, Vertical takeoff

aircraft, Noise prediction, Noise reduction, Ducts, AcousticThe propulsion systems of current and future V/STOL linings, Computer programsvehicles can be defined as combinations of free-air propellers,shrouded propellers, variable pitch fans, fixed pitch fans, tilt

71

A computer program is presented which allows a user to technique to: (a) discretize the structure into equivalentmake performance and far-field acoustic noise predictions systems of finite elements, (b) select the displacement modelfor free-air propellers, variable pitch fans with inlet guide most closely approximating the real case, (c) derive elementvanes, variable pitch fans with outlet guide vanes, fixed pitch and assemblage stiffness and inertia properties, and finallyfans, helicopter rotors, tilt rotors, fixed pitch lift vanes with (d) form the matrix equations of motion and the resultingremote, integral, and tip-turbine drives, and variables pitch eigenvalue problems. The stiffness and Inertia properties arelift fans with remote and integral drives. Noise prediction evaluated by expressing the potential and kinetic energies ofmethodology for drive engines, single stream and coaxial the element (or the assemblage) in terms of nodal displace-jet;, and gearboxes are also included, as well as noise reduc- ments. Detailed numerical examples are presented to illus-tion and performance losses of partly sonic inlets and duct trate the applicability and effectiveness of the analysis and toacoustic treatment. A description of the program, detailed investigate the dynamic characteristics of suspension bridgesinstructions for its use, required inputs, and sample cases are with widely different properties.presented.

77-35777-355 Structural Analysis and Retrofitting of ExistingPavement Response to Aircraft Dynamic Loads. Highway Bridges Subjected to Strong Motion SeinnicVolume Ill. Compendium LoadingR.H. Ledbetter R.R. Robinson, E. Privitzer, A. Longinow and K.H.Army Engineer Waterways Experiment Station, ChuVicksburg, MS., Rept. No. WES-TR-S-75-11-Vol-3, lIT Research Inst., Chicago, IL., Rept. No. IITRI-FAA-RD-74-39-3, 94 pp (June 1976) (See also Vol J6320-FR, FHWA/RD-75-94, 330 pp (May 1975)1, AD-A016 450) PB-255 299/0GAAD-A028 378/8GA

Key Words: Bridges, Earthquake resistant structures, SeismicKey Words: Aircraft, Landing, Pavements, Runways response

Instrumented aircraft were used to apply static and dynamic The objective of this project was to determine cost effectiveloads to instrumented pavement structures (both flexible and means for modifying existing Intermediate size bridges so asrigid) at the National Aviation Facilities Experimental Center to better withstand the damaging effects of Intense earth-(NAFEC), Atlantic City, NJ. quake ground motions. Research studies were performed to

identify and define, through structural analysis, practicaltechniques and criteria for retrofitting the bridges selectedduring the program. The need for retrofitting is based on the

BRIDGES philosophy that damage should be limited so that collapsedoes not occur and traffic can be restored after minimumrepairs. Seven different bridge structures were selectedthroughout the United States in high risk seismic regions.Seismic loads were determined for each bridge based on the

Dynamic Analyses of Suspension Bridge Structures soil conditions and seismicity at its site. A simplified analysisand Some Related Topics procedure was defined during the project and validated byA M. Abdel-Ghaffar selective comparisons with nonlinear response analyses.Ph.D. Thesis, California Inst. of Technology, 460 pp(1976)UM 76-23, 378 BUILDING

Key Words: Suspension bridges, Linear theories, Finiteelement technique 77-358

The thesis is divided into two parts. The first part develops Modal Control of Multistory Structures

a method of dynamic analysis for vertical, torsional and C.R. Martin and T.T. Soonglateral free vibrations of suspension bridges, based on lin- Dept. of Engrg. Sci,, State Univ. of New York atearized theory and the finite-element approach. The method Buffalo, Buffalo, NY, ASCE J. Engr. Mech Div., 102involves two distinct steps- Ill specification of the potential (EM4), pp 613-623 (Aug 1976) 4 figs, 7 refsand kinetic energies of the vibrating members of the con-tinuous structure, leading to derivation of the equations ofmotion by Hamilton's Principle, (2) use of the finite-element Key Words. Multistory buildings, Modal control technique

72

In this paper the design of active control systems for civil Key Words: Locks (waterways), Excavations, Groundengineering structures is considered using model control motionconcepts. The theory of modal control is briefly preentedwith respect to the control of linear structures where the The U.S. Army Corps of Engineers recently replaced the oldcontrol objective is to direct changes of its specific dynamic Bankhead Lock on the Black Warrior River, Alabama, with amodes and stiffness. larger lock. The new lock required an approach canal ex-

cavated by the removal of more than 5 million cu yd ofrock. Preliminary test shots and an exploratory excavationcontract of 0.35 million cu yd established blasting and

CONSTRUCTION monitoring techniques that assured that vibration and settle.(Also see No. 346) ment of the old lock would not be excessive. Shots were

monitored with particle velocity pickups on and adjacent tothe lock. A relationship between peak particle velocity andscaled distance was checked on the basis of these data after

77-359 normalization for structure behavior, coupling, and trans-Vibratory Compaction of Bituminous Concrete missivity. Recommendations from the test shot program werePavements used In specifications for the exploratory excavation con

C.D. Burns tract. The exploratory excavation, with about 50 presplit andArmy Engineer Waterways Experiment Station, production shots, served as a second testing program forrefining the previous observations and conclusions, accordingVicksburg, MS., Rept No WES-MP-S-76-10, 61 pp to various blesthold arrays and time delays.(June 1976)AD-A026 843/3GA

HELICOPTERSKey Words: Vibrators (machinery), Compaction equipment, HAlso see Nos. 312,352,353,354)

Compacting, Concretes, Pavements

This study was conducted to evaluate the effectiveness ofvibratory rollers In the compaction of hot-mix asphaltic 77.361concrete and rubberized-tar concrete to satisfy the needs of In.Flight Far.Field Measurement of Helicopterthe Air Force. The study consisted of over-laying an existing Impulsve Noiseheavy gear load test section at the U.S. Army EngineerWaterways Experiment Station, which consisted of rigid and F.H. Schmitz and D.A. Boxwellflexible pavements, with asphaltic concrete and rubberized- U.S. Army Air Mobility R&D Laboratory, Moffetttar concrete pavements. The overlay pavements were com- Field, CA., J. Amer. Helicopter Soc., 21 (4), pp 2-16pacted with two selected vibratory rollers, a Buffalo-Bomag (Oct 1976) 13 figs, 13 refsBW210-A and a Dynapac CC-50A. A conventional steel-wheeled static roller and a pneumatic-tired static roller werealso used for comparison. Variables included in the study Key Words: Helicopter noise, Measurement techniqueswere roller weight, frequency and amplitude of vibration,number of roller passes, type of roller (vibratory or static), A new and highly successful method of collecting far-fieldtype of foundation, and type and thickness of overlay acoustic data radiated by helicopters in forward flight haspavements, been developed, utilizing a quiet aircraft flying in formation

ahead of the subject helicopter, The teed aircraft, flown asan acoustic probe, was equipped with tape-recording equip-ment and an external microphone. Spatial orientation of the

EARTH helicopter with respect to the monitoring aircraft was achiev-ed through visual flight reference. Far-field acoustic datadefining the impulsive noise radiation characteristics of theUH-AH helicopter during high-speed flight and partial-power

77-360 descents have been gathered with this technique. ThreeReview and Analysis of Blasting and Vibrations at distinct types of impulsive waveforms have been identifiedBankhead Lock and correlated with helicopter steady operating conditions.

R.J. LuttonArmy Engineer Waterways Experiment Station,Vicksburg, MS , Rept No WES-TR-S-76-6, 86 pp 77.362(June 1976) AH-I Helicopter Vibration Levels for Stub WingAD-A026 735/1GA Mounted Equipment

73

D. Welford and J.S. Boland, III Driving point mechanical impedance measurements were used^ ,.. lAiS:IC r elnent and Engrg. ab to determine the dynamic response of the human head toR hn sinusoidal vbraton n the frequency rnge b|teen 0 HzRedstone Arsenal, AL., Rept. No. RG-76-30, 147 pp and 5000 liz at excitation levels of 0.98 m/s2 and 3.4 ms 2.

(Sept 1975) Because of the low excitation levels, the weight of the headAD-A026 825/OGA was sufficient to couple the head to the vibration source.

The response of the head to vibration can be simulated by aKey Words: Helicopters, Wing stores two degree-of-freedom, mass-excited system consisting of a

series connection of a small driving mass, a damper, a spring" and damper In parallel end a large final mass. Parameter

This report presents measured translational and angular ad der In prenre ss. Par eacceleration inputs to stub wing mounted equipment on the values, derived by computer techniques, suggest that the largeAH-1 helicopter, In this case the wing mounted electro- mass represents the total head, the small mass the tissue Inoptical systems were the stabilized platform airborne laser contact with the vibration input and the spring the skull

system and the stabilized mirror airborne laser system. The stiffness.

data are presented in the form of power spectral density,amplitude versus frequency, and oscillograph charts.

ISOLATION(Also see No. 401)

HUMAN

77-36577-363 User Manual for Air Bag Restraint Program ABAG 19An Experimental Study of Package Cushioning for R.H. Dufortthe Human Head Vizex, Inc., Buffalo, NY., Rept. No., VCR-76-1,Y.K. Liu and K.B. Chandran DOT-HS-801 929,, 124 pp (July 1976)

Biomechanics Lab.,, School of Medicine and Engrg., PB-256 11517GA

Tulane Univ., New Orleans, LA., J, AppI. Mech.,Trans. ASME, 43 (3), pp 469-474 (Sept 1976) Key Words: Air bags (safety restilint systems), Mathematical7 figs, 16 refs models, Computer programs

Sponsored by the National Institute of Health and The major tasks covered by this report are the collection,the National Science Foundation interpretation, clarification and documentation of the

accumulated but undescribed changes evolved by the variousKey Words: Head (anatomy), Impact shock, Isolation, users of a computer simulation of air beg behavior.

Experimental data

An experiment was performed to determine the containeracceleration and pressure distribution in a Plexiglass cylinder, 77-366filled either with water or 3 percent set.gelatin, and impacted Vibration Study of Radar Receiver/Transnitter M 163against a well. This experiment serves to quantitatively Vulcan Air Defense Systemvalidate a theoretical model simulating an one-dimensional J.H. Wilandclosed-head impact given earlier. The experiments showedimportant differences between the theoretical and experi-mental pressure measurements. FA-TR-76017, 35 pp (Mar 1976)

AD-A027 448/OGA

77-364 Key Words: Equipment mounts, Vibration control

Dynamic Properties of the Human Head This report describes a laboratory vibration study conductedJ.B. Smith and C.W. Suggs on the Receiver/Transmitter unit of the M163 Vulcan AirDept. of Biological and Agricultural Engrg ,, North Defense System. Field reports indicated an undesirableCarolina State Univ , Raleigh, NC, J., Sound Vib., amount of vibration in the unit during vehicle operation on

paved roads, resulting in excessive maintenance requirements.48 (1), pp 35-4 3 (Sept 8, 1976) 4 figs, 10 refs The laboratory study determined the equipment's critical

frequencies likely to be excited by vehicle operation, and theKey Words: Head (anatomy), Mechanical impedance, Vibra- associated mode shapes. The primary vibration problem istion response identified as insufficient support structure for the microwave

chassis.

74

MATERIAL HANDLING The problems which confront the engineer wishing to moni-(Also see Nos. 346, 347) tor the condition of rotating machinery are first of all dis-

cussed in general terms. The subject of comparative diagnos-tics is then introduced and illustrated by reference to case

77-367 .,istories of both bearing and gear failure.

A Systematic Approach to Noise Reduction of ArmyFork Lift TrucksE.F. Ellingson METAL WORKING AND FORMINGAdvanced Technology Center, Allis-Chalmers Corp.,SAE Paper No. 760600, 20 pp, 45 figs

77-370Key Words: Hoists, Trucks, Noise reduction Experimental and Analytical Investigation of Self.

Excited Chatter in Metal CuttingThis report discusses work performed under government N. Saravanja-Fabriscontract to study and reduce noise emission of Army 6000lb Ph.D. Thesis, Illinois Inst. of Technology, 153 ppcapacity fork lift trucks. Sound levels of stock fork lift (1976)trucks were studied in detail Including operator exposure,Contributing noise sources were identified and studied, and UM 76-23,498several abatement techniques were established for eachsource. Key Words: Metal working, Cutting, Chatter, Self-excited

vibrations

Chatter In metal cutting Is a nonlinear self excited vibration77.368 of the limit cycle type. The mechanism that causes it is not

Noise Control on a Heavy-Duty Mobile Crane yet fully understood. The first pert of this Investigation IsJ.R. Bernhagen concerned with the analysis of chatter from the viewpoint of

Harnischfeger Corp., SAE Paper No. 760601, 16 pp, the describing function.

7 figs, 26 refs

Key Words: Cranes (hoists), Noise reduction OFF-ROAD VEHICLES

The intent of this paper is to illustrate how simple testingtechniques and application of elementary noise concepts 77-371helped attenuate the noise in the operator cabs of a heavy-duty mobile crane to within the 8 hour/90 dBA limit estab- Asesment of Occupational Noie Exposre andlished by the Occupational Health and Safety Act of 1970. Associated Hearing Damage Risk for Agricultural

EmployeesJ.D. Harris, B.J. Lindgren and R.L. Mann

MECHANICAL J.1. Case Company, Milwaukee, WI., SAE Paper No.

(Also see Nos 255,406) 760673, 12 pp,9 figs, 8 refs

Key Words: Agricultural machinery, Noise measurement,77.369 Measurement techniques, Noise tolerance

Vibration Analysis as an Aid to the Detection and This paper describes background information, measurementDiagnosis of Faults in Rotating Machinery procedures, data analysis techniques, and results pertainingR M. Stewart to assessment of agricultural employee occupational noise

Inst. of Sound and Vibration Res, Univ. of South- exposure. The criteria on which the proposed OSHA 85db(A) 16 hour general industry noise regulation is based hasampton, Southampton, England, "Conference on been applied to predict hearing damage risk. Analysis of

Vibrations in Rotating Machinery", The Inst. of 237 man days of measured noise exposures shows thatMech Engrs , Univ. of Cambridge, Sept 15-17, 197b, annual occupational noise exposure must be established topp 223-229, 5 figs, 13 refs assess hearing damage risk. No evidence was found that

agricultural employees are exposed to noise levels/durationswhich exceed the criteria for annual and lifetime noise

Key Words: Rotating structures, Gears, Bearings, Machinery, which the rop a geal ind usDiagostc tehniuesexposure on which the proposed OSHA general industry

Diagnostic techniques regulation is based.

75

77-372 the analytical model a linear parametric representation isFederal Regulation of Noise in Agricultural and Off- obtained by fitting to time-response data over the normal

Highway Equipment operating band of the engine.

F.A. Van AttaQuinnipiac College, SAE Paper No. 760674, 3 pp,4 refs RAIL

Key Words: Agricultural machinery, Noise reduction,Regulations 77-375

This paper seeks to answer questions pertinent to Federal DYNALIST il, A Computer Program for Stabilityregulations of noise in agricultural and off-highway equip- and Dynamic Response Analysis of Rail Vehiclement. It discusses the statutes under which authority lies, Systems. Volume 1. Technical Reportand whether or not the regulations actually accomplish the T.K. Hasselman, A. Bronowicki and G.C. Hartpurposes for which they are intended.

Transportation Systems Center, Cambridge, MA,Rept. No. DOT-TSC-FRA-74-14.1, FRA/ORD-75/22.1,118 pp (Feb 1975)

PUMPS, TURBINES, FANS, PB-256 046/4GACOMPRESSORS(Also see No. 307) Key Words: Interaction: rail-wheal, Railroad cars, Computer

programs

A methodology and a computer program, DYNALIST II,77-373 have been developed for computlng the response of railIdentification of an Automotive Gas Turbine. Part 1. vehicle systems to sinusoldal or stationery random railFrequency Response Estimation Irregularities. A modal synthesis procedure Is used whichP.E. Wellstead and DJ. Nuske permits the modeling of subsystems or components by

Control Systems Centre, Univ. of Manchester Inst. of partial modal representation using complex elgenvectors.Complex elganvectors represent the amplitude and phase

Science and Technology, Sackville St., Manchester characteristics of rail vehicle systems which occur as a resultM60 1QD, England, Intl. J. Control, 24 (3), pp 297- of wheel-rail interaction, heavy damping In the suspension309 (Sept 1976) 8 figs, 11 refs system and rotating machinery. Both vertical and lateral

motion are handled by the program which allows up toKey Words: System identification, Motor vehicles, Gas twenty.f lve component and fifty system degrees of freedom.turbines, Frequency response

A two-stage automotive gas turbine is subjected to an Identi-fication exercise. The system is studied using digital fre- 77.376quency response estimation procedures, and locally linearized Acoustic Impacts of BART: Interim Service Findingstransfer functions are obtained which span the normal Metropolitan Transportation Commission, Berkeley,working region of the engine. CA., Bolt Beranek and Newman, Inc,, Cambridge,

4! MA., Rept. No. MTC-TM-16-4-76, 93 pp (Mar 1976)

PB-254 966/5GA77-374

Identification of an Automotive Gas Turbine. Part 2. Key Words: Rapid transit railways, Sound measurement,Parameter Fitting Vibration measurement

D.J Nuske and P.E. Wellstead The report documents the findings and methodologies de-Control Systems Centre, Univ. of Manchester Inst. of veloped during a study of BART sound and vibration levels.Science and Technology, Sackville St., Manchester The findings focus on: delineation of impacted regions, majorM60 1QD, England, Intl. J. Control, 24 (3), pp 311 - factors affecting BART-generated sound, prototype vs. opera-

E n I tlonal sound levels, BART vs. other transportation sound324 (Sept 1976) 9 figs, 10 refs sources and BART-generated vibration levels. BART-generat-

ad sound levels were derived from direct wayside measure-Key Words: System identification, Motor vehicles, Gas ments and indirectly from on-board recording of sound levelsturbines throughout the BART system. Ambient community souna

Using insight gained from a frequency-domain identification levels were based on predictive techniques verified by fieldexperiment a simple analytical model is formulated for an measurements.

automotive gas turbine. With a structural model based upon

76

77-377 of motion and design considerations are discussed. The in-

Lateral Dynamics of a Tracked Air-Cushion Vehide fluence of various physical parameters on the response of theP.J. Davis and R .J. HawKs pipe and the restraint, and the effects of using different finiteelement models are considered. Specific emphasis is directedGenercI Electric Co., Philadelphia, PA., High-Speed to the verification of the accuracy of the solutions obtained

Ground Transp. J., 10 (2), pp 135-146 (Summer usingenergybelancechecks.

1976) 5 figs, 9 refs

Key Words: Tracked vehicles, Ground effect machines,Dynamic stability, Equations of motion Energy Absorbers Used Against Impact Loading

Linearized equations of motion for a TACV are developed in T. Kukkolathe three degraes-of.freedom of lateral displacement, yaw Nuclear Power Project Group, Imatran Voima Osa-angle, and body roll angle. Air cushion forces and the aero- keyhti6, Helsinki, Finland, Nucl., Engr. Des., 37 (3),dynamic loads on the body are included by means of stability pp 407-412 (June 1976) 9 figs, 2 refsderivatives. The lateral dynamic stability of the vehicle isanalyzed by the use of root-locus diagrams. Key Words: Nuclear reactor containment, Pipes (tubes),

Energy absorption

REACTORS In the WWER440 reactor the primary piping consists of sixhorizontal loops going rapidly from the pressure vessel, eachloop having a horizontal steam generator. In this reactor type

the relatively long primary piping with many curved sections77378 requires special attention in order to successfully eliminateAeroelastic Modeling of Large Wind Turbines the consequences of the design basis accident. EmergencyP.P. Friedmann supports are located in appropriate places to restrict the

Mech. and Struct. Dept., University of California, Los movements of the pipe. Under normal conditions there Is

Angeles, CA., J. Amer. Helicopter Soc.,21 (4), pp 17- a gap of some centimeters between the pipe and a support sothat in the pipe can be deformed freely under changing loads.

27 (Oct 1976) 6 figs, 28 refs This paper deals with those energy-absorbing structures usedat the Lovilsa Nuclear Power Plant for protection against

Key Words: Turbine blades, Mathematical models, Equations impact loading. Places and circumstances where energy.of motion absorbing structures are employed ore specified. Develop-

ment and design of impact absorber elements are discussedA set of coupled flap-lag-torsional equations of motion for and impact tests are described.

a single wind turbine blade are derived in a general, non-linear, partial differential form. These equations are suitablefor determining the aeroelastic stability or response of largewind turbine blades. Methods for solving the equations 77.381together with some possible simplification of the equations The Use of Energy Absorbers to Protect Structuresare discussed. Finally, the formulation of the complete rotor- Against Impact Loadingtower aeroelastic problem is considered in general terms. P. Hernalsteen and L.C. Leblois

TRACTIONEL, Societe de Traction et d'Electricite,B-1040 Brussels, Belgium, Nucl. Engr. Des., 37 (3),

77.379 pp 373-406 (June 1976) 57 figs, 29 refsOn Finite Element Analysis of Pipe Whip ProblemsS.M. Ma and K.J. Bathe Key Words: Nuclear reactor containment, Pipes (tubes),Bechtel Power Corp., San Francisco, CA 94119, Nucl. Energy absorption, Experimental data

Engr. Des , 37 (3), pp 413-429 (June 1976) 14 figs, When a flying missile impacts a fixed structure, the interface

10 ref s loading is dependent on the deformation characteristics of

both impacting and impacted bodies. The following "energyKey Words: Nuclear reactor containment, Pipes (tubes), absorption" materials and processes have been experimen-Mathematical models, Finite element technique tally investigated, with an aim at pipe whipping restraint

application for nuclear power plants; plastic extension ofThe nonlinear dynamic finite element solution of pipe whip austenitic stainless steel rods, plastic compression of copperproblems is presentod. The finite element modeling used, the bumpers, and punching of lightweight concrete structures.step-by-step incremental solution of the nonlinear equations

77

77.382 In attempting to identify the dynamic characteristics of alarge medium-speed diesel engine using pseudo-random bi-nary sequence (p.r.b.s.) and cross-correlation techniques

D. Dini and L. Lazzeri inconsistent results were obtained. The cause of these is due

Comitato Nazionale Per I'Energia Nucleare, Divisione to the sampled-data nature of the diesel engine operation and

Sicurezza e Controlli, Rome, Italy, Nucl. Engr. Des., the difficulties were compounded by having to run so large

37 (3), pp 361-371 (June 1976) 11 figs, 8 refs an engine under closed-loop control conditions for safetyreasons. Part 1 of this paper identifies the source of suchproblems arising from the use of such techniques for investi-

Key Words: Nuclear reactor containment, Pipes (tubes), gating sampled-data systems and develops a variation of theMathematical models conventional theory which allows a wide class of both open

and closed-loop sarnpled systems to be identified. Part 2 ofThe dynamic model used to describe the pipe and the re- the paper is concerned with the application of this theory tostraint to predict their behavior during a shock phenomenon extensive meawurements of a large engine.such as an earthquake Is described. The pipe is simulated bymeans of a series of finite elements, having a complex elasto-plastic behavior. Each element can be subdivided in to anumber of subelements whose flexibilities are computed and 77-385internally condensed at the element level. Dynamic Measurements of a Large Diesel Engine

Using P.R.B.S. Techniques. Part 2. Instrumentation,Experimental Techniques and Results

77.383 J.O. Flower and G.P. WindettPipe Whip Analysis for Nuclear Reactor Applications School of Applied Sciences, Univ. of Sussex, Falmer,N. Bisconti, L. Lazzeri and P.P. Strona Brighton, BN1 9QT, England, Intl. J. Control, 24 (3),SAIGE, Societa de Architettura Industriale per gli pp 393-404 (Sept 1976) 11 figs, 6 refsImpianti di Generazione di Energia, 1-16151 Genova-Sampierdarena, Italy, NucI. Engr. Des., 37 (3), pp Key Words: Diesel engines, Dynamic properties, Measuring347-360 (June 1976) 9 figs, 14 refs Instruments, Measurement techniques

The work reported here in Part 2 concerns the application ofComuey W orms the methods, discussed in Part 1, to obtain dynamic measure-Computer programs ments from a large turbo-charged diesel engine. The small

perturbations of this approach do not significantly affectThe main problems encountered In pipe whip analysis are normal engine running and can be well correlated withdiscussd. Problems discussed are breakage locations, and nomlegeruigadcnbewlcreatd ihdrcued. ro b i ssed avarious linear models of the engine. The engine was a 750

b.h.p. rated unit designed specifically as the prime moverfor electrical generation on a ship. In these experiments,however, the power produced was absorbed by a water-brake. For safety reasons It was not permitted to run theengine in other than a closed-loop configuratlon'.RECIPROCATING MACHINE

77.384 77-386

Dynamic Measurements of a Large Diesel Engine On Piston Slap in Reciprocating Machinery

Using P.R.B.S. Techniques. Part 1. Development Y. Fujimoto, T. Suzuki and Y. Ochiai

of Theory for Cloed-Loop Sampled Systems Dept. of Mech. Engrg , Tottori Univ., Japan, "Conf.J 0. Flower and G.P. Windett on Vibrations in Rotating Machinery," The Inst. of

School of Applied Sciences, Univ of Sussex, Falmer, Mech. Engrs., Univ. of Cambridge, Sept 15-17, 1976,

Brighton, BN1 9QT, England, Intl. J. Control, 245-253, 13 figs, 14 refs(3), pp 379-392 (Sept 1976) 8 figs, 7 refs

Key Words: Reciprocating engines, Pistons, Lubrication

Key Words: System identification, Diesel engines

78

Piston slap is a phenomenon when a piston Impacts the 77-389cylinder wall, traveling across the small cylinder clearance Mini RPV Engine Noise Reductionspace. during the course of its vertical movement along the R.M. Shimovetz and D.L. Smithcylinder axis in reciprocating machinery. Such lateral motionof the piston is caused by the change of the direction of the Air Force Flight Dynamics Lab., Wright-Pattersonsidewise force acting on the piston. The dynamics of lateral AFB, OH, Rept. No. AFFDL-TR-76-28,, 97 pp (Marand rotational motions of the piston were studied both 1976)theoretically and experimentally. Theoretical results obtained AD-A027 638/6GAby numerical integration predict the continuous movementsof the piston. The analysis Includes the effect of the oil filmformed on the piston skirt surface as well as the frictions at Key Words: Engine noise, Reciprocating engines, Noise

the rings. Experiments were carried out on e reciprocating reduction

compressor. Four miniature inductive displacement trans-ducers oi the piston skirt made it possible to determine both The purpose of this effort was to Investigate the reduction in

the transverse ad rotational movements of the piston con- radiated acoustic noise associated with two types of engines

tinJously. Comparison of calculated and measured move- considered for power plants in small (75-100 lb.) remotely

ments showed the importance of oil cushion effect on piston piloted vehicles (Mini RPV) in the class of the Preaelre andstap. Calere Aircraft. The two engines considered are approximate-

ly 5 HP; the first a rotary combustion (RC), the second a twostroke cycle reciprocating (P). The sound pressure levelswere recorded using a semicircle arrangement of micro.

77-387 phones In a free field and with various engine noise reductionThe Effect of Pistons in Diesel Engine Noise Gen- devices Installed. The engines were rotated such that aeration spherical definition of the acoustic pressures were made.

M. RohrleOstfildern, MTZ Motortech. Z., 37 (7/8), pp 277-282(July/Aug 1976) 9 figs, 16 rufs 77-390(In German) An Integrated Approach to Measurement of Exhaust

and Intake NoiseKey Words: Diesel engines, Engine noise, Noise measurement, D.A. Olson, D.C. Flanders, and L.J. ErikssonMeasurement techniques Corporate Research Dept., Nelson Industries, Inc.,

SAE Paper No 760602, 12 pp, 21 figs, 10 refsA method for measuripg diesel engine noise caused by pistonhitting the cylindei wall is described. Key Words: Engine noise, Noise measurement

An integrated approach for measurement of exhaust and77-388 intake noise during a single test is presented and typical

results discussed. Internal pressure waveforms as well asInterior Noise Problems of Small Diesel-Powered external radiated noise are included for various intake andBuses exhaust systems.G.L. MarlotteH. L. Blachford, Inc , SAE Paper No. 760603,12 pp,15 figs, 12 refs ROAD

(Also see Nos 285, 293, 348, 388)Key Words: Buses (vehicles), Diesel engines, Engine noise

This paper discusses interior noise problems characteristicof the combination of diesel engines with typical small busgeometries. One noise problem is the potential for low-f re- Biomechanics of Trauma ins Childrenquency resonances. Natural air vibration modes for typical G. Rutersmall bus interior dimensions are calculated and shown to Automobiltech Z., 78 (7/8), pp 339-342 (July/give frequencies that can be stimulated by noise and vibration Aug 1976) 7 figs, 28 r-ef 5at engine firing frequencies. Measurem3nts of noise and (In German)vibration at resonance are given. Other noise problems areassociated with internal front engine covers, and leaks andlow transmission loss paths in the console area. Some prac- Key Words: Collision research (automotive), Human factorstical suggestions, with examples, are given to reduce interior engineeringnoise.

79

L7&

This paper reports about an international meeting of the The mechanical properties of vehicles are investigated as aInternational Research Committee on the Biokinetics of basis for favorable crash energy management solutions.Impacts in Lyon. September 1974. Important parameters follow the similarity law of Cauchy.,

Different concepts of inter-vehicular compatibility arediscussed. The tolerable closing speeds depending on the massratio and accelerations of colliding vehicles are calculated

77-392 using realistic data of deformation forces, human loadPerformance Evaluation of Child Test Dummies. tolerances and different restraint system characteristics.Volume 1. Technical Report Engine effects and measures to reduce engine aggressivenessM.P. Shah and V.G. Radovich are investigated. Possible solutions for accident types other

than frontal collisions are briefly discussed.Transportation Research Center of Ohio,, East Liber-ty, OH, Rept. No. DOT-HS-801-921, 261 pp (May1976) (See also Vol 2. PB-255 119)PB-255 118/2GA

Radio Telemetry Applied to Tire VibrationsKey Words: Collision research (automotive), Anthropo- W.F. Reiter, Jr. and A.C. Eberhardtmorphic dummies, Experimental data Center for Acoustical Studies, N. Carolina State

Univ., Rayleigh, NC, SAE Paper No. 760745, 12 pp,Commerically available three- and six-year-old child dummies 10 figs,10 refswere evaluated for their anthropometric measurements anddynamic response characteristics in pendulum impact testsand simulated crashes in representative automobile-child Key Words: Automobile tires, Vibration measurement,seat restraint environments. Simulated crashes included 20- Measurement techniquesand 30-mile-per-hour frontal and 20-mph side Impacts on This paper discusses the industrial radio telemetry system forautomobile and specially designed bench seats. use in typical automotive engineering measurements. Con-

siderations of transducer type and measurement class (staticor dynamic) are discussed in terms of selection of a telemetry

77.393 system. relemetry specifications and their interpretationsDummies. are discussed along with characteristics and limitations of

Performance Evaluation of Child Test direct/FM and AM/FM systems. As an example applicationVolume II. Appendix the measurement of In-service tire vibrations with radioM.P. Shah and V.G. Radovich telemetry is presented. The example Includes selectionTransportation Research Center of Ohio, East Liber. and calibration of the telemetry system and presentationty,, OH, Rept No. DOT-HS-801-922, 269 pp (May of typical telemetered tire vibration data.

1976) (See also Vol. 1, PB-255 118)PB-255 119/OGA

77-396Key Words: Collision research (automotive), Anthropomor- Proceedings of a Symposium on Commercial Vehiclephic dummies, Experimental data Braking and Handling Held on May 5.7, 1975 at

Ann Arbor, MichiganThis Appendix contains samples of the computer calculated a Arbor, Mihi adata packages produced for each test. The digital date corre- Highway Safety Research Inst, Michigan Univ.,lates with descriptions and data parameters discussed in the Ann Arbor, MI, Rept. No. UM-HSRI-PF-75-6, 599main body of the report. pp (1975)

PB-255 985/4GA

77.394 Key Words: Motor vehicles, Tires, Brakes (motion arresters),

Development of Crash Energy Management Solutions Dynamic response, Proceedings

H. Danckert , The symposium was to provide a review of the state of theVolkswagenwerk AG, Germany, SAE Paper No. art in commercial vehicle braking and handling. It was also760793, 20 pp, 19 figs 27 refs designed to provide a forum in which members of various

research organizations, industrial operations, and governmentagencies involved with the topic could communicate ideasand concerns over problems related to measuring and/or

simulating the braking and directional response of commer-cial vehicles.

80

L!

T7.397 investigated. Both rail-car structural configurations and im-Tentative Road Roughness Criteria Based Upon pact absorption devices were studied. This final report issued

Vehicle Performance under the crashworthiness effort covers the development of

B.E. Quinn and S.R. Kelly analytical tools to pedict passenger threat - environmentSchool of Mechnr., Pu e Uduring collision; criteria for predicting passenger injury dueSchool of Mech. Engrg., Purdue Univ., Lafayette, IN, to train collisions; an application of injury criteria andRept. No. FHWA-RD-75-3, 107 pp (June 1975) analytic models to predict passenger injuries resulting fromPB-254 809/7GA collisions of trains that represent existing construction types;

a preliminary investigation of applying impact absorptiondevices to transit vehicles; a design study of car structural

Key Words: Pavement roughness, Tire characteristics configurations for improved impact energy management and

Pavement roughness is defined In terms of two parameters a review of engineering standards for Urban Rail Car Crash.

of an equation used to represent the roughness spectrum. worthiness.

The relationship between these parameters and vertical tireforce, passenger acceleration and lateral tire force is thendeveloped. This is the third of three reports on pavement 77400roughness and vehicle dynamics studies under this contract. Vehicle Test Facilities at Aberdeen Proving Ground

Army Test and Evaluation Command, Aberdeen

Proving Ground, MD, Rept. No. TOP-i-1-011, 61 pp77-398 (Mar 17,1976)Study of Aerodynamic Drag Reduction on a Full- AD-A027 035/5GAScale Tractor-TrailerL. L. Steers and L.C Montoya Key Words: Military vehicles, Tracked vehicles, Test facilitiesDryden Flight Res Ctr., NASA, Edwards AFB, CA,Rept. No DOT-TSC-OST-76-13, 57 pp (Apr 1976) The report describes Aberdeen Proving Ground facilities forPB-254 571/3GA testing wheeled and tracked vehicles including vehicular

weapon systems. Includes photographs and drawings showingtest course dimensions and characteristics. Does not cover

Key Words: Articulated vehicles, Aerodynamic loads, Dy- equipment and instrumentation used on the courses nornamic tests laboratory facilities except for climatic test chambers.

Aerodynamic drag tests were performed on a tractor-trailercombination using the coast-down method on a smooth,nearly level runway. The tests Included an investigation of 77401drag reduction obtained with add-on devices that are com- Control of Vehicle Dynamics Considering Guidewaymercially available or under development. The tests coveredti ctor-trailer speeds ranging from approximately 35 to 65 Design Criteriamiles per hour and included fuel consumption measurements. S.L. Graham and D.A. Hullender

Vought Corp., Dallas, TX, AIAA J., 14 (8), pp 1022-1025 (Aug 1976) 4 figs, 5 refs

77.3"9 Key Words: Interaction: vehicle-guideway, SuspensionAn Asaesuent of the Crash",orthinesa of Existing systems (vehicles)

Urban Rail Vehicles. Vr' e Ill. Train-Collision

Model Users Manual The ride quality of a transportation vehicle is dependent onD J Se(lal the effectiveness of the suspension system to filter out

Calspan Corp., Buffalo, NY, Rept. No DOT-TSC- vibration due to guideway roughness and other sources suchUMTA-75-21.111, UMTA-MA-06 0025-75-16.111, 66 as wind gusts. To meet ride comfort specifications without

unreasonable stroke requirements, it is sometimes necessarypp (Nov 1975) to compensate the suspension design with some form ofPB-254 695/OGA either active or passive control. This paper addresses the sub-

ject of relating the tradeoff in ride comfort for suspensionKey Words: Crashworthiness, Collision research (railroad), stroke to specific guideway profile design tolerances andEnergy absorption constraints. Specifically, approximate lower bounds of RMS

acceleration for any vehicle suspension system as a functionThe crashworthiness of existing urban rail vehicles (passenger of RMS suspension stroke are computed for specific guide-cars) and the feasibility of improvements in this area were way profile elevation variation constraints and construction

surveying accuracies.

81

77402 Vibration problems are encountered on different types of

Truck Noise Viii: The Determination of the Prac- rotating equipment in Process Industries which can lead to

tical Noise Control Retrofitting of Pre-1970 Truck very high economic losses due to the need to shut down forand Coach Models rectification. This paper outlines some of the more difficult

problems met in recent years. Brief accounts are given ofO.J. Bullard and G.M. Shaffer the symptoms of looseness and how they were identified on

GMC Truck and Coach Div, General Motors Corp., a centrifugal fan and a centrifugal compressor, the identifi-

Pontiac, MI, Rept. No DOT-TSC-OST-75-51, 120 pp cation of vibration induced by rubbing of rotating elements

(June 1976) (PB-241 114) on stationary parts, the cause of a modulating noise levelfrom a gearbox driving an induced draught fan, problemsarising from machinery being mounted on flexible bedplatesand structures, and effects of bearing oil film on rotor

Key Words: Motor vehicle noise, Trucks, Noise reduction vibration.

A retrofit noise package was selected for four representativeGMC vehicles to achieve optimum noise reduction. Theselection of this material came from commercially available 77405items submitted by various component suppliers. A new Vibrations in Cracked Shaftssystem of nolse-source Isolation was developed in order to T A. Henry and B.E. Okah-Avaeevaluate the vendor-supplied material. Dept. of Engrg., Simon Engrg. Laboratories, Univ.

of Manchester,, England, "Conf. on Vibrations inRotating Machinery," The Inst. of Mech. Engrs.,

77403 Univ. of Cambridge, Sept. 15-17, 1976, pp 15-19,Investigation on Vibration Characteristics of Un. 6 figs, 3 refsspring Vehicles with Pneumatic TyresA. Owzar Key Words: Diagnostic techniques, Shafts, Rotors

Teheran, Iran, Automobil-tech. Z.,, 78 (9), pp 377-380 (Sept 1976) 8 figs, 10 refs The vibrations of a rotor are examined using analogue com-

(In German) puter simulation. A crack is introduced as a reduction inshaft stiffness when the shaft deflects so as to open thecrack. The interaction of gravity and the crack excites

Key Words: Tractors, Seats, Vibration measurement resonances at main and sub critical speeds. Out-of-balancecombines with the crack to change the vibrations at the criti-

The purpose of this experiment was to examine the vibration cal speed only in certain situations. The study, confirmed bycharacteristics of tractors. Practical measurements on the experimental data, shows that the gravity effect dominates invibration characteristics of various vehicles were made to horizontal shafts balanced to normal standards and that theevaluate the effect of vibration on drivers, growth of a crack would be detected from the vibrations at

the main and sub critical speeds.

ROTORS(Also see Nos 251, 332) 77406

Vibration in Rotating Machinery: MalfunctionDiagnosis - Art & Science

77-404 E. DownhamVibration Problems on Rotating Machinery in the The Univ. of Aston, Birmingham, England, "Conf.Chemical Industry on Vibrations in Rotating Machinery," The Inst.J.B. Erskine and C.W. Reeves of Mech. Engrs., Univ of Cambridge, Sept. 15-17,Noise and Vibration Section, Agricultural Div., 1976, pp 1-6,2 figs, 3 refsICI Ltd., Billingham Cleveland TS23 1 LD, England,"Conf. on Vibrations in Rotating Machinery," The Key Words: Diagnostic techniques, Rotating structures,Inst of Mech. Engrs., Univ. of Cambridge, Sept. Machinery15-17, 1976, pp 209-214, 7 figs, 8 refs

Key Words: Rotors, Compressors, Diagnostic techniques

82

The generalized approach to vibration malfunction diagnosis critical speed may be obtained and the probability of insta-of rotating machinery is shown to require a sound under- bility may be calculated. The effects of influence coeffi-standing of fundamental theoretical concepts associated with cients measurement errors on the balancing weight determin-machine and machine element dynamics, an appreciation of ation are investigated for the balancing of flexible rotors.the nature of the dynamic forces and Instabilities whichexcite vibration in various types of machinery, together withthe ability to plan concise experiements to obtain practicaldata regarding the cause or likelihood of failure. The wide 77409variety of vibration phenomena which result in malfunction Modern Influence Coefficient Techniques for Multi-is illustrated by case studies on various types of industrial plane Rotor Balaning in the Factory, Tes CeU andmachinery. Field

R.H. BadgleyMechanical Technology, Inc., Latham, NY, "Conf.

77.407 , on Vibrations in Rotating Machinery," The Inst.On the Determination of the Influence Coefficients of Mech. Engrs., Univ. of Cambridge, Sept. 15-17,in Rotor Balancing, Using Linear Regression Analysis 1976, pp 201-207, 2 figs, 8 refsL. LarssonBalancing and Vibration Control Dept., ASEA, Key Words: Rotors, Balancing techniques, Influence co-Sweden, "Conf. on Vibrations in Rotating Machine- efficient matrixry," The Inst. of Mech. Engrs., Univ. of Cambridge,Sept. 15-17, 1976, pp 93-98, 4 refs Several modern techniques have been developed for perform-

-n multiplane-multispead balancing by the influence coeffi-cient procedure. These techniques include the use of digital

Key Words: Rotors, Balancing techniques, Influence coeffi- minicomputers to control the performance of multiple low-cient method cost spin-up stands in high-production factory environments,

as well as the use of remotely accessed time-sharing computerBalancing rotating bodies, using the conventional Influence systems for field balancing and occasional user applications.coefficient method does not always lead to a satisfactory Significant improvements in balance quality are now achiev-result. The reason Is often uncertainties In the response from able, together with manufacturing cost reductions. New mea-a set of attached test wvights which have been superimposed chinery Is now being designed to take advantage of this newby internal, unknown alterations of the unbalance dlstribu- capability.tion caused by e.g. setting of the rotor windings or mechani-cal parts. In this paper, the emphasis Is put on the determin-ation of the influence coefficients, and a method for statis-tical evaluation is presented. T:ie reliability of the so ob- 77-410tained influence coefficients is investigated on rigid and Systematic Combination of Experiments and Dataflexible rotors balanced in a test pit, and the application ofone influence matrix for the balancing of other similar rotors Processng in Balancing of Ilexihie Rotorsis studied. J. Drechsler

Technische Universitat, Berlin, Germany, "Conf.on Vibrations in Rotating '.,achinery," The Inst.

77408 of Mech. Engrs., Univ. of C.nb-idge, Sept. 15-17,

Error Analysis t, Vibration or Rotor/Bearing SyLem 1976, pp 129-132, 7 refs

T. lwatsuboKob Iwasb o b, JKey Words:, Rotors, Balancing techniques, Influence coeffi-Kobe Univ , Kobe, Japan, "Conf. on Vibrations in cient matrixRotating Machinery,' The Inst. of Mech Engrs,Univ of Cambridge, Sept. 15-17, 1976, pp 87-92, Under the asumption that an approximate influence coeffi-6 figs, 11 refs cient matrix is available, the balancing process of a flexible

rotor can be considerably accelerated by a consequent com-bination of experiments and data processing , which yields

Key or"s" Rotor-bearing systems, Error analysis, Balancing balancing weights and an improved influence coefficienttechniques, Stability analysis matrix after each trial run. Even if a trial run has to be inter-

rupted because of excessive vibration amplitudes, the pro-This paper deals with tho error analysis of the vibration of psed mecyields balaning wigtich williede the

roto berin sytem. Ifthestaistcalproprtis o erors posed methoo yields balancing weights, which will reduce therotor bearing systems. If the statistical properties of errors vibration amplitudes of the explored speed range and will not(i.e., mean values and standard deviations) are krown, the amplify the vibration level in the remaining unknown opera-mean . lije and standard deviation of unbalance response and tion speed range.

83

77411 Key Words: Rotors, Balancing techniquesModal Resolution and Balancing of SynchronousMobal utio n ai dl Baoators oft ynon ouosmrvv In the last few years the volume of literature concerningVibrations in Flexible Rotors with Non-Conservatie flexible rotor balancing has grown by leaps end bounds.Cros Coupling Most authors concentrate on the theoretical aspects so thatH.F. Black and S.M. Nuttall the intioduction of ne, practical method is progressing slow-Heriot-Watt Univ., Edinburgh, Scotland, "Conf. on ly, and attempts at putting theory Into practice are oftenVibrations in Rotating Machinery," The Inst. of quickly abandoned due to a lack of clear instructions fromMech. Engrs., Univ. of Cambridge, Sept. 15-17, 1976, theoreticians. This paper Is Intended to provide practicalM r Uguidelines for proper selection and application of flexiblepp 151-158, 4 figs, 14 refs rotor balancing techniques.

Key Words: Rotor-bearing systems, Balancing techniques,Mathematical models

77414The paper examines the response to mass unbalance of a The Whirl Modes of Vibration of a Crankshaftgeneric rotor In hydrodynamic bearings. The non-conserva- D. Hodgettstive equations of motion are separated by bi-normal ortho- School of Automotive Studies, Cranfield Inst. ofgonalization. Modal resolution of unbalance response Isdiscussed in relation to balancing. Examples of modal resolu- Technology, England, "Conf. on Vibrations intion and balancing calculations are given. Rotating Machinery," The Inst, of Mech. Engrs.,

Univ. of Cambridge, Sept., 15-17, 1976, pp 255-261,4 figs, 3 refs

77.412A Method for Using the Free Shaft Modes in Rotor Key Words: Crankshaft, Whirling, Natural frequencies, Mode

Balancing shapes

J.W. Lund A theoretical three-dimensional model has been developed toTechnical Univ. of Denmark, Lyngby, Denmark, provide the frequencies and modal shapes of the coupled"Coi.f. on Vibrations in Rotating Machinery," The whirling, axial and torsional modes of vibration of a crank-

shaft. A satisfactory correlation of the measured and theo.Inst. of Mech. Engrs., Univ of Cambridge, Sept. 15- retical frequencies of the vibrations was obtained to., fre-17, 1976, pp 65-71,4 figs, 9 refs quencies below 500Hz. The possibility of uting the model

to predict the amplitudes of vibration is oiscussed.Key Words: Shafts, Mathematical models, Modal analysis,Balancing techniques

From a correlation of measured and calculated natural fre- 77415quencies and modes It is assumed that a representative analy- Dynamnic Stiffness Matrix Approach for Rotor.tical model can be established for the free shaft. With the bearing System Analysisshaft installed and running at some selected speed the unbal- N.F. Rieger, C B. Thomas and W.W. Walterance response amplitudes are measured at several pointsalong the rotor after which a resolution of the response into Rochester Inst. of Technology, Rochester, NY,its free mode components is performed. The unknown "Cont on Vibrations in Rotating Machinery," Thebearing reactions, including damping forces, are evaluated Inst. of Mech. Engrs., Univ. of Cambridge,, Sept. 15-either by trial weight runs, one run per bearing, or, in case of 17, 1976, pp 187-193, 8 figs 15 refsjust two bearings, from a force balance. Elimination of thebearing reacztions leads to the determintion of the massunbalance in terms of its modal expansion, and correspond- Key .Words: Rotor-bearing systems, Stiffness methods,

ing correction weights are then readily found. Critical speeds, Unbalanced mass response, Stabilityanalysis, Computer programs

The dynamic stiffness matrix concept is preseted as a gen-

77413 oral technique for calculating the unbalance response, criticalsPractice of Flexible Rotor Balancing speeds (either damped or undamped) and stability thresholdrspeed of a flexible rotor in damped flexible supports. A linear

A. Giers elastic rotor system in eight-speed dependent coefficientCarl Schenck AG, Darmstadt, W. Germany, "Conf bearings is considered. A computer code based on this con-on Vibrations in Rotating Machinery," The Inst of cept is described.Mech Enqrs, Univ of Cambridge, Sept 15-17, 1976,pp 33-42, 16 figs, 15 refs

I

77416 Key Words: Modal analysis, Rotating structures, Cantilevers!! A Method of Stabilizing Extemally-Preeurized Gas

A Mo Sabi A discussion of the modal analysis of a rotating flexible struc-a. ture is presented. Specifically this work is concerned with the

D.J. Woodford free vibration analysis of a rotating cantilever structure.

Central Electricity Res. Labs., Leatherhead, England, A brief discussion of the formulation of the equations of"Conf. on Vibrations in Rotating Machinery," The motion for the system is included to aid in understanding theInst. of Mech. Engrs., Univ. of Cambridge, Sept. 15- origin of the various terms appearing in these equations.

17, 1976, pp 111-115, 6 figs, 4 refs

Key Words: Rotors, Whirling, Stabilization 77419The Modal Interpretation of the Vibration of a

Rotors supported In externally-pressurized gas journal- The Moa terpreat

bearings are subject to an instability known as self-excited Damped Rotating Shafttranslational whirl. The onset of whirl limits the maximum A.G. Parkinsonspeed to which a rotor may be run. This paper shows how Mech. Engrg. Dept., University College, London,the upper bound to the mass parameter may be Increased England, "Conf. on Vibrations in Rotating Machine-by means of a simple modification to the rotor's surface. ry,' The Inst Mech. Engrs., Univ. of Cambridge,

Sept. 15-17, 176, pp 261-269, 4 figs, 9 refs

77417 Key Words: Rotors, Shafts, Modal analysisInvestigations into Load Dependent Vibrations ofInvestigations P into r Loa redent T Vb.rato of Much of the theoretical progress In rotor dynamics has beenthe High Pressure Rotor on Large Turbo-Generators made in terms of modal analysis. Practical applications ofS.H. Greathead and P. Bastow the theory, however, have rosulted in many, often misleading,

CEGB NE Region Scientific Services Dept., Otley references to principal modes, damped modes and "twisted"

Road, Harrogate, England, "Conf. on Vibrations in modes. The relationships between these various concepts and

Rotating Machinery," The Inst. of Mech. Engrs., the restrictions to their use are discussed. The intention isto remind the practical vibration engineer that no method ofdiagnosis and correction, such as balancing, Is likely to be

7 figs, 8 refs successful in an industrial context unless the actual shaft

vibration is correctly Interpreted.Key Words: Turbomachinery, Rotors, Stability analysis

Vibrational instability of the high pressure rotor on certaintypes of large turbo-generator operated by the C.E.G.B. has 77420sometimes developed above a particular load level. When this A Strategy for Investigating the Linear Dynamics ofinstability arises very high non-synchronous vibration of the a Rotor in Bearingsrotor and bearings occurs necessitating load restrictions, I. Fawzy and R.E.D. BishopIn the past, this type of non-synchronour vibration was Dept, of Mech. Des, Faculty of Engrg., Univ. ofusually attributed to bearing instability but the presentinvestigations show that the instability originates mainly in Cairo, Egypt, "Conf. on Vibrations in Rotatingthe steam glands at the high pressue inlet to. the machine. Machinery," The Inst. of Mech. Engrs., Univ. ofThe results show that the rotor instability is a self-excited Cambridge, Sept. 15-17, 1976, pp 239-244, 1 fig , 7vibration due to steam forces generated in these glands refsresulting from the steam leakage flow through them andthat the rotor to gland alignment is an important parametercontrolling the load at which instability occurs. Key Words: Rotors, Rotor-bearing systems, Linear analysis,

Modal analysis

Linear modal analysis of rotor dynamics necessitates the77-418 making of simplifying assumptions when the rotor is sup-

Modal Analysis of Rotating Flexible Structures ported in bearings that are not 'ideal', i.e. when it is not

R.M. Laurenson housed in free, pinned, clamped or sliding bearings. A theo-retical method is outlined by which this difficulty can be

McDonnell Douglas Astronautics Co., East St. Louis, overcome. The rotor/bearing system is treated as the non-

MO, AIAA J-, 14 (10), pp 1444-1450 (Oct 1976) conservative system it is and a -odified form of linear modal9 figs, 14 refs analysis is employed.

85

77421 77424Vibrations of a Rotating Asymmetric Shaft Carrying Rotor Vibration: The Choice of Optimal JournalTwo Disks, Supported in Asymmetric Bearings BearingsL. Forrai D. MorrisonDept. of Mechanics, Technical Univ. of Heavy Y-ARD Ltd., Consulting Engineers, Glasgow, Scot-Industry, Miskolc, Hungary, "Conf. on Vibrations land, "Conf. on Vibrations in Rotating Machinery,"in Rotating Machinery," The Inst. of Mech. Engrs., The Inst. of Mech. Engrs., Univ. of Cambridge, Sept.Univ. of Cambridge, Sept. 15-17, 1976, pp 43-48, 15-17,1976, pp7-14,5figs, 5refs2 figs, 7 refs

Key Words: Rotors, Vibration response, Optimization,Key Words: Shafts, Rotors, Perturbation theory Journal bearings

Parametrically excited lateral vibrations of a rotating asym- A wide body of experience is now sccmulsting on themetric light shaft carrying two disks, supported in ssymmetri- prediction of the vibratory behavior of rotor systems. Suchcally flexible bearings, are analyzed by the approximate predictions are in reasonable agreement with practical be-perturbtion-varlation method of Hsu. The solutions of the havior and there Is a close relationship with the general formsequations of motion give simple explicit expressions for of behavior which are intrinsic in some early simple pre-stability boundaries provided the shaft asymmetry is sma;l, dictions. The relationship of these behavioral patterns to theThere is no limitation on the degree of bearing asymmetry. absolute bearing clearance Is noted as being particularlyThree-dimensional diagram shows the unstable speed-regions important and the numerical values involved are such thatfor a rotor system with asymmetries of both the cross-section It appears possible to arrive at optimal bearing clearances forand the bearing. par~lcular rotors, which are clearly not impractical. It is

noted that wide ranges of bearing clearance ratio are in com-mon use and hence that there is no real physical barrier tovarying the clearances if this is vibrationally desirable. A

77422 simple (to carry out) method of determining an optimum

A Quick, Graphical Way to Analyze Rotor Whirl clearance for a rotor Is demonstrated by an example and the

H.D. Nelson and D.A. Glasgow derivation is given.

Arizona State Univ., Tempe, AZ, Mach. Des, &8(23),pp 124-130 (Oct 7,1976) 6 figs

77425Key Words: Rotor-bearing systems, Rotors, Whirling, Mathe- Characteristic Vibrations of Flexural Rotors inmatical models, Graphic methods Journal Bearings

E. Pollmann and H. SchwerdtfegrComplete design studies of rotor/bearing systems usually Kraftwerk Union AG, Mulheim-Ruhr, W. Germany,

involve complex digital computer programs. But pre- "Cof on ions i Rottn Mahiery,

liminary studies often can be based on greatly simplified "Conf on Vibrations in Rotating Machinery," Themath models. Here is a set of desigrn curves that take some of Inst. of Mech. Engrs., Univ. of Cambridge, Sept.the drudgery out or rotor analysis. 15-17, 1976, pp 21-26, 14 figs, 3 refs

Key Words: Journal bearings, Shafts, Rotors, Vibration77-423 response

A Practical Vibration Primer The vibration behavior of an electrical generator was exam-Part 6. Critical Speeds and Mode Shapes med by considering similar geometric and dynamic condi-C. Jackson tions on a model with various types of bearings. When run-Monsanto Co., Texas City, TX, Hydrocarbon Proces- ning through the first and second critical speeds, the unbal-

g 5ance vibrations show the large amplitudes typical for slendersing, 55 (10), pp 141-146 (Oct 1976) 12 figs rotors. In order to examine stability behavior, the damping

of a shock induced vibration was ascertained. In a wideKey Words: Rotors, Critical speeds, Mode shapes range of rotation speeds low frequency dependent on the

oil supply pressure occured with a limited amplitude.All turboequipment operates at certain speeds where theforcing or self-exciting frequencies are sympathetic (atsynchronous or harmonics) to the resonances of a rotor,bearing, support system. When this resonance occurs at a fi-nite operating speed, that s;::ed is referred to as a criticalspeed. This basic article describes these phenomena.

86

L!

77426 The advances that have been made in the past several years in

Experimental Studies on the Shaft Dynamics of the area of rotor-bearing analysis is changing the basic designsfor an Improved Surveillace procedure of the major industrial compressor manufacturers.Large Turbogenerators fThe capability to calculate undamped critical speeds, forced

A. Azzoni, A. Clapis, R. Gariboldi, G. Lapini, G. response of damped rotor-bearing system, damped eigen-Possa, and T. Rossini values for linearized stability, and time transient responseCentro Informazioni Studi Esperienze, Segrate, for large amplitude whirl and shock loading studies is nowMilano, Italy, "Conf. on Vibrations in Rotating available to conduct design reviews of proposed rotor sys-

Machinery," The Inst. of Mech. Engrs , Univ of tems. Some of the problems that still exist in rotor dynamics

Cambridge, Sept. 15-17, 1976,, pp 173-179, 13 figs, analysis are discussed.

1 ref

Key Words: Shafts, Turbomachinery, Testing techniques, 77429Test equipment Numerical Analysis of Free Vibrations of Damped

Rotating StructuresThe paper presents the main results of some experimental K.K. Guptastudies on the shaft dynamics of two ENEL large turbo- Jet Propulsion Lab., Calif. Inst. of Technology,generators, i.e. a 70 MWe unit at the Emilla power station(Piacnzas and a 320 MWe unit at the TurbigoLante power Pasadena, CA.,, "Conf. on Vibrations in Rotating

station. The tests were performed within the frame of a Machinery," The Inst. of Mech. Engrs., Univ. ofresearch programme aimed at improving the continuous Cambridge, Sept. 15-17, 1976, pp 27-31,, 1 fig,automatic surveillance techniques. 7 ref s

Key Words: Rotating structures, Damped structures, Numer-

77427 ical analysis, Free vibrations, Finite element technique,

Normal Mode Interaction in Self-Excited Rotor Component mode synthesis, Rotors

Vibrations This paper is concerned with the efficient numerical solutionE. Brommundt of damped and undamped free vibration problems of rotatingMechanikzentrum, Lehrstuhl A f(ir Mechdnik, Tech- structures. While structural discretization is achieved by thenische Universitt Braunschweig, Germany,, "Conf. finite element method, the associated eigenproblem solutionon Vibrations in Rotating Machinery," The Inst. of is effected by a combined Sturm sequence aid inverseiteration technique that enables the computation of a fewMech. Engrs , Univ. of Cambridge, Sept. 15-17, required roots only without having to compute any other.1976, pp 105-109, 5 figs, 5 refs For structures of complex configurations, a modal synthesis

technique is also presented, which is based on appropriate

Key Words: Rotors, Stability analysis, Normal modes combinations of eigenproblem solution of various structuralcomponents. Such numerical procedures are general in

When a rotor gets unstable, its oscillations are closely related nature, which fully exploit matrix sparsity inherent in finiteto one of Its normal modes. At different parameter sets, element discretizations, and prove to be most efficient fore.g., different speeds, different normal modes can act. When the vibration analysis of any damped rotating structure, suchthese simple motions, too, get unstable, combinatory mo- as rotating machineries, helicopter and turbine blades, soin-tions with interaction of the normal modes can occur, ning space stations, among others.

77428 77-430The Impact of Rotor Dynamics Analysis on Ad. The Soluti,,n of Vibration Problems with Simplevanced Turbo Compressor Design ModelsR G. Kirk E Kramer, R Nordman, and H.D. KlementIngersoll-Rand Co., Turbo Products Div., Phillipsburg, Technische Hochschule Darmstadt, German FederalN J,, "Conf on Vibrations in Rotating Machinery," Republic,, "Conf. on Vibrations in Rotating Machine-The Inst. of Mech Ergrs., Univ of Cambridge, ry," The Inst. of Mech. Engrs , Univ of Cambridge,

Sept 15-17, 1976, pp 139-150, 7 figs,, 16 refs Sept 15-17, 1976, PP 79-86, 12 figs

Key Words: Shafts, Machinery, Elastic foundations, NaturalKey Words: Rotor-bearing systems, Compressors, Turbines, frequencies

Stability, Design techniques

87

It is usual today to calculate the vibratioris of turbo.sats in 77433most cases with separate models for shaft and foundation, Large Amplitude Whids of Rotorsbecause the necessary combined calculation is expensive G. Capriz and A. Larattaand time-wasting. In some cases more simple models aresufficient, whose behavior is also Illustrative for the under- Istituto di Elaborazione della Informazione C.N.R.standing of larger systems. This will be shown with the Pisa, Italy, "Conf. on Vibrations in Rotating Machine-example of a machine shaft, mounted on a block founda- ry," The Inst. of Mech. Engrs., Univ. of Cambridge,tion. Sept. 15-17, 1976, pp 49-52, 4 refs

Key Words: Shafts, Lubrication77431Dynamic Be7av3our of a Simple Rotor with a Croso large amplitude vibrations are considered in shafts rotatingSecyinact Cros on lubricated bearings, for a certain class of lubricant re-Sec¢tional Cra~k sponse; the behavior of the shaft is described in differentR. Gasch ranges of speed of rotation. It is shown finally that the analy.Technical University, Berlin, Germany, "Conf. on sis applies partly also to more general cases of lubricant

Vibrations in Rotating Machinery," The Inst. of response (e.g., short bearingswith cavitating oil film).

Mech. Engrs., Univ. of Cambridge, Sept. 15-17, 1976,pp 123-128,9 figs, 2 refs

SATELLITEKey Words: Rotors, Cracked media, Stability, Whirling (See No. 418)

In this paper the dynamic behavior of a rotor consisting ofa disc on a flexible shaft with a cross-sectional crack is anal-yzed. The (non-linear) equations of motion are derived by SHIPreplacing the crack by a simple mechanism. Experiments have (Also see No. 337)shown, that the deflection-properties of this model comevery close to that of the shaft with a real crack. Some ,esultsof an analog-computer-study are given for the influence of 77434weight and unbalance on stability and whirl. Two Media Low Pressure Transient Measurements

H.E. Krachman, L.G. Lindsey, and W.F. CaplanDevelopmental Sciences, Inc., City of Industry, CA,

77432 ISA Transactions, 16 (1), pp 17-21 (1976), 15 figsEffect of Damping on the Flutter Boundary ofRotating Systems Key Words: Hydrofoil craft, Ships, SlammingK, HuseyinUniv. of Waterloo, Ontario, Canada, "Conf. on Pressure transients weremeasured on the bow section of a

Vibrations in Rotating Machinery," The Inst. of model surface effect ship while investigating the bow slam-

Mech Engrs., Univ. of Cambridge, Sept. 15-17, 1976, ming phenomenon.

pp 133-137, 4 figs, 9 refs

Key Words: Rotors, Flutter, Vibration damping 77435Nature of Ship Collisions within Ports

The effect of internal and external damping on the diver- J.E. Warwick and A.L. Andersongence and flutter boundaries of rotating systems under the Res. and Tech. Div., Todd Shipyards Corp., Galve-influence of a conservative force (e.g. axial load) is studiedin general terms on a comparative basis with the correspon- ston, TX, Rapt. No. NMRC-336-40000-R, MA-RD-ding undamped system. Certain lower and upper bounds as 920-76060,48 pp (Apr 1976)well as convexity properties are established. As an example, PB-255 304/8GAthe flexible rotating shaft subjected to axial load is analyzed.

Key Words: Nuclear powered ships, Collision research (ships)

88

The report is a summary of the nature of ship collisions 77438which have occurred in ports or while in the protected Experimental Study on Oscillatory Combustion inwaters enroute to port and is based on information obtained Solid-IPropelaj t Motorsfor ship operations in seven representative U.S. ports over afive year period. In addition to the collisions which have T. Koreki, I. Aoki, K. Shirota, Y. Toda, and K.occurred, port traffic information for the same time period KurataniIs included. Nissan Motor Co., Ltd., Tokyo, Japan, J. Space-

craft and Rockets, 13 (9), pp 534-539 (Sept 1976)5 figs, 9 refs

77-436Structural Analysis of Aircraft Impact on a Nuclear Key Words: Solid rocket propellants, Combustion excitation

Powered Ship The relationship between the longitudinal mode oscillatory

R. Dietrich combustion phenomena in solid-propellant motors and the* Institut f Or Analagentechnik, Gesellschaft f Or Kernen- motor configuration Is experimentally examined using small- ergieverwertung in Schiffbau und Schiffahrt mbH., test motors with various grain configurations.

D-2054 Geesthacht-Tesperhude, Germany, Nucl.Engr. Des., 37 (3), pp 333-345 (June 1976) 19 figs,16 refs 77439

Vibration of a Flexible Spacecraft with MomentumKey Words: Collision research (ships), Collision research (air- Exchange Controllerscraft), Nuclear powered ship%, Dynamic response, Finite J.R. Canavinelement technique Ph.D. Thesis, Univ. of California, Los Angeles, 172 pp

As part of a general safety analysis, the reliability against (1976)structural damage due to an aircraft crash on a nuclear UM 76-22,179powered ship Is evaluated. This structural analysis is an aidin safety design. It is assumed that a Phantom military Key Words: Spacecraft, Vibration response, Rotating struc-jet-fighter hits a nuclear powered ship. The total reaction erd

force due to such an aircraft impact on a rigid barrier isspecified In the guidelines of the Rea ktorSicherheitskommIs- Significant problems are presented in the vibration andsion (German Safety Advisory Committee) for pressurized rotation anlysis of spacecraft with distributed structurl

water reactors. This paper Investigates the aircraft impact flexibility and momentum exchange controllers. Theseon the collision barrier at the side of the ship. systems exhibit gyroscopic coupling which depends on the

rotor speed and orientation, which must remain explicitIn the analysis as control variables. An investigation is made

SPACECRAFT into floating reference frames In order to allow first ordervibration analysis In the presence of large system rotations.(Also see No. 325)

77437 77-440

Appendage Modal Coordinate Truncation Criteria Digital Program for Computing the Transfer Functionin Hybrid Coordinate Dynamic Analysis and Frequency Response of the Fluidgenics Regula.P. Likins, Y. Ohkami, and C. Wong tor (CCD-PROG-024)Columbia Univ., New York, NY, J. Spacecraft and C.A. RothRockets, 13 (10), pp 611-617 (Oct 1976) 1 fig, General Dynamics/Convair, San Diego, CA, Rept. No.9 refs GDC.DDE66-014, SAMSO-TR-76-173, 78 pp (Mar

Key Words: Spacecraft, ")ynamic analysis 1966)AD-A027 769/9GA

Sever) alternative candidates for hybrid and normal coor-dinate truncation criteria are offered for consideration, Key Words: Pressure regulators, Frequency response, Com-

interpreted, and illustrated by example. Two of the criteria puter programs, Missile componentsare based on system eigenvalues, two on system eigenvectors,and three on measures of controllability and observability.No definitive basis is provided for selection among the seven

criteria, but comparisons are offered.

89

The Fluidgenics Pressure Regulator Program was developed TURBOMACHINERYto evaluate the linearized, steady-state transfer function (Also see Nos. 306,308, 310, 378,417,4261associated with the Fluidgenics pressure regulators used onthe ATLAS LAUNCH VEHICLES. The regulator analysiswas made in conjunction with the 5 cps longitudinal oscil-lation stability analysis. The important quantities generated 77443by the program are the steady-state gains, damping ratios, Dynamic Response of Cavitating Turbonachinesand break frequencies for the transfer function; the transfer S.-L. Ngfunction in equation form, and the magnitudes and phase Ph.D. Thesis, California Inst. of Technology, 113 ppangles as a function of input frequency. (1976)

UM 76-23,385

STR UCTU RAL Key Words: Turbomachinery, Cavitation

Stimulated by the pogo instability encountered In manyliquid propellent rockets, the dynamic behavior of cvi-

77441 tating inducers Is the subject of the present thesis. An experi-Prediction of Noise Distribution in Various Enclo- mental facility where the upstream and downstream flowssures from Free-Field Measurements of a cavitating inducer could be perturbed was constructed

and tested. The upstream and downstream pressure and massA.G. Galaitsis and W.N. Patterson flow fluctuations were measured. Matrices representing theBolt Beranek and Newman, Inc., Cambridge, MA transfer functions across the inducer pump were calculated02138, J. Acoust Soc. Amer., &0 (4), pp 848-856 from these measurements and from the hydraulic system(Oct 1976) 8 figs, 11 refs characteristics for two impellers in various states of cavita-

tion. The transfer matrices when plotted against the pertur-bing frequency showed significant departure from steady

Key Words: Rooms, Tunnels, Enclosures, Noise prediction state or quasi-steady predictions especially at higher frequen.cies.

The image theory method is used to predict sound pressurelevels generated by an acoustic source of known strength inregular rooms, tunnels, and flat rooms. For regular rooms,the results are Identical to those of the well-established 77-444diffuse field theory. For tunnels and flat rooms, the predic- Acceptance Criteria for Compresors and Turbinestions are verified by direct measurements taken in under-ground mines, in the Process Industry Based on the Response to

Deliberate UnbalanceP.E. Simmons

77442 ICI Petrochemicals Div., Wilton, Middlesbrough,England, "Conf. on Vibrations in Rotating Machine-Seianic Design Methods for Military Faclities - ry," The Inst. of Mech. Engrs., Univ. of Cambridge,

Preliminary Recommendations Sept. 15-17,, 1976, pp 73.78,4 figsW.K. StockdaleConst. Engrg. Res. Lab., U.S. Army, Champaign, I L, Key Words: Turbomachinery, Unbalanced mass responseRept. No. CERL-IR-M-184, 80 pp (June 1976)AD-A027 384/7GA A generation of large turbo-machines for the process indus-

tries is emerging in which it is difficult and may be unde-Key Words: Se~smic design, Military facilities sirable to achieve the degree of separation between the

operating speed range and the lateral critical speeds that isThis report presents preliminary recommendations for the commonly specified. Alternative criteria are suggested whichmethods of structural analysis to be used in the design of are based on the response of the rotor to deliberate unbal-critical facilities on military installations. The recommended ance.dynamic analysis methods are described and discussed, andexamples are presented which illustrate the elastic andinelastic response spectra methods. It is recommendedthat dynamic methods be used in all areas where the ex-pected ground acceleration exceeds 0.10 g.

90

77-445 USEFUL APPLICATIONVibrational Problems in Modem Power Station PlantW.G.R. Davies, A.W. Lees, I.W. Mayes, J.H. Worsfold,and F.J.P. Crampton 77447Mech. Sci. Section, CEGB Midlands Region, Scien- Vibroflotation and Terra-Probe Comparisontific Services Dept., Radcliffe-on-Soar, Nottingham, R.E. Brown and A.J. GlennEngland, "Conf., on Vibrations in Rotating Machine- Law Engrg. Testing Co., Washington, D.C., ASCE J.ry," The Inst. of Mech. Engrs., Univ. of Cambridge, Geotech. Engr. Div., 102 (GT 10), pp 1059-1072Sept. 15-17, 1976, pp 215-222, 7 figs, 3 refs (Oct 1976)

Key Words: Turbomachinery, Power plants (facilities), Key Words: Vibratory techniques, Compacting, SoilsDiagnostic techniques

The most widely used techniqucs for deep vibratory corn-The paper describes the combined application of theoretical pection of cohesionless soils are Vibroflotation and Terra-and standard experimental techniques to the diagnosis of Probe. The Vibroflotatlon process consists of penetrating theoperational problems on a modern power generation plant, soil with a horizontally vibrating probe and simultaneouslyin the first example, thermal changes In the relative heights saturating and vibrating the soil as the probe is extracted.of bearing pedestals were shown to cause one bearing of a The Terra-Probe process consists of driving and retracting alarge turbogenerator to become unloaded leading to insta- 30-in. (760-mm) diem open-ended steel pipe with verticalbility and excessive vibration. At another location, a large vibrations induced by a vibratory pile driver. Three testfan was discovered to have critical speeds within the running patterns with different probe spacings were compacted withspeed range due to gyrodynamic forces on the overhung 100-hp Vibroflot probes and three test patterns were com-impeller. pacted by Terra-Probe. The test results are discussed In this

paper.,

77.446Vibration Excited by Lateral Forces Caused by theLeakage Flow in Thermal TurbomachineryK. UrlichsTechnische Universitat Munchen, Lehrstuhl f. Tech-nische Kraftanlagen,, Arcisstrasse 21, D-8000 Mun-chen 2, BRD, Germany, Ing. Arch., 45 (3), pp 193-208 (1976) 12 figs, 13 refs(in German)

Key Words: Vibration excitation, Turbomachinery

Self excited vibrations of turbo rotors can be caused bylateral forces proportional to the displacement, which resultat the turbine stages from the leakage flow between the rotorand the casing. For the calculation of these forces the vari-able clearances are represented by a set of stream tubeswith various cross sections. So the local mass flux and thepressure distribution in the clearance follows by means ofone-dimensional empirical relations, taking into accountthe circumferential velocity at the influx. Measurementson a test turbine gave good agreement with calculated results.

91

AUTHOR INDEX

Abdel-Ghaffar, A.M........ 356 Chandran, K.B .... ....... 363 Giers, A ................ 413Adi Murthy, N.D. . 320 Chen, P.J ........... .... 265 Glasgow, D.A ............. 422Aksu, G .. .... .338 Chien, C.F...... ........ 241 Glenn, A.J ............... 447Ali, R.........38 Chou, P.C............ 297 Grabowski, B .......... 251

Allaire, P.E .... .......... 306 Chu, K.H.........357 Graham, S.L ............. 401Alwar, R.S .............. 320 Chung, T.J . .325 Greathead, S.H ............ 417Anand, G.V ............. 258 Clapis, A ............... 426 Greif, R ................. 341Anderson, A. L........... 435 Clarke, J.D .... .......... 337 Gunter, E.J .............. 306Aoki, I................ 438 Confer, V.J .............. 252 Gunzburger, M.D .......... 239Arnoldi, R A .......... .. 307 Crampton, F.J.P .......... 445 Gupta, K.K..............429Au-Yang, M.K ........ .... 327 Danckert, H. ............ 394 Hall, M..... ........... 234Ayres, DJ ............ .. 281 Davies, W.G. R........ ... 445 Halliwell, D.G ............ 309Azzoni, A....... ....... 426 Davis, P.J ............... 377 Harris, J.D ............... 371Badgley, R.H .... ........ 409 De Hoog, F.R ............ 235 Hart, G.C ............... 375Bai, K.J . 279 Delph,T.J ............ .273 Hashin, Z ............... 283Baig, M.L. ... 329 Dib, G.M ....... .... ... 334 Hassab, J.C ............. 233Bannister, R. H .305 Dietrich, R .. . . .. 436 Hassel man, T. K........ 275,375Barrett, L.E...........306 Dini, D. . .382 Hawks, R J .............. 377Bastow, P. ......... 417 DoiI, W..........282 Hegdahl, T...........266,267Bates, C L ............... 321 Downham, E... ...... ... 406 268, 269, 270Bathe, K.J .. ............ 379 Drechsler, J........... 410 Hemmings, R.C ........... 304Becker, RJ .. .... 344 Dubey, RK . . ....330 Henry, T.A.............. 405Bennett,, B.E . 232 Dufort, R.H .. . . .. .. 365 Hernalsteen, P............. 381Berge, B ... .......... 299 Eberhardt, A.C ... ... 395 Herrmann, G ......... 232, 273Bernhagen,J R.. .. ...... 368 Eidson, R L.. .. ... .. 325 Hilber, H.M ............. 249Be7ler, P ............. 322 Ellingson, E.F............367 Hino, M .......... ...... 280Bigret, R . .. .. 308 Eriksson, .. ............. 390 Hodgetts, D .............. 414Bisconti, N . . .... 383 Erskine, J B .... ....... 404 Houghton, J. R............ 256Bishop, R E D. .420 Ewins, D.J........310 Hud, G C ............... 255Black, H.F..... . .. 411 Fagerlund, A.C,......324 Hughes, T.J R............ 249Blackstock, D.T .. 345 Faw, y,l.........420 Hugus, G.D ............. 286Boland, J S., Ill.I.........362 Flanders, D C. . 390 Hullender, D.A ............ 401Boxwell, D A.. ....... 36 Fletcher, NK H.... 257 Hundal, M S ............. 343Bragg, E E....... . 288 Flis, W J. ......... 297 Huseyin, K .............. 432Brommundt, E .. 427 Flower, J.. 0.384,385 Ivey, E S........ ...... 260Bronowicki, A . 375 Ford, M.B.........264 Iwan, W.D .............. 339Brown, R E. .. 447 Forrai,, ....... 421 lwatsubo, T. ............ 408Bullard, 0 J. . 402 Friedmann, PP . . 378 Jackson, C.........423Burdess, J.S...... 284 Fujimoto, Y. . 386 Jacquot,R G3... . 245Burns, C D . .359 Fukuoka, H .. 272 Jenissen, A 266, 267,Canavin, J.R 439 Galaitsis, A G.......441 268, 269, 270Caplan, W F .. 434 Gale, J G .. 300 Johnson, A F.......294Capri/, G . . 433 Gariboldi, R 426 Johnson, G,.R.......250Carta, F 0 307 Gasch, R .431 Junghans, R........347

92

Kaliski, S .............. 277 Mann, R.L.............. 371 Quinlan, P.M ............. 326Kamil, H ........ ....... 263 Marlotte, G.L ............. 388 Radovich, V.G ......... 392,393Kaper, B ................ 243 Martin, C.R .............. 358 Ramamurti,V ............ 328Kaul, R.K....... ....... 273 Maunder, L ............. 284 Reed, C ............... 323Keller, J.B ............. 236 Mayes, L.W ............ 445 Reeves, C.W ............. 404Kelly, S.R ...... ........ 397 Meirovitch, L ... ......... 254 Reifsnider, K.L........... 274King, W.F., III .......... 291 Miles, A.W ............... 290 Reiter, W .F., Jr ........... 395Kinney, W .A ............. 261 Minagawa, S ............. 271 Riegel, C ................ 301Kirk, R.G ............... 428 Mirza, S ................ 333 Rieger, N.F....... . . ... 415Kisliakov, S.D ........... 335 Mitchell, G.C ............ 337 Robinson, R.R ........... 357Kleinstein, G.G ......... . 239 Mittendorf, S.C ........... 341 Rohrle, M ............... 387Klement, H.D .. .... 430 Moeller, T.L ............ 339 Romilly, N .............. 240Kluwick, A ............. 242 Molina, M.A ...... ..... 303 Rossini,T ............... A26Koltzsch, P ........ ... . 347 Montoya, L.C ............ 398 Roth, C.A ............... 440Koreki, T ..... ..... ... 438 Morrison, D .............. 424 Rudny, D F .............. 346Kisloff, D ..... ......... 262 M Uller, J ... ............ 317 Ruter, G ......... ...... 391Koval, L.R ......... . .. 351 Murata, S ... ........... 311 Sachs, H.K .............. 348Krachman, H.E ........... 434 Murthy, V.R ... ...... ... 312 St. Hilaire, A ............. 307Kramer, E .... ....... .. 430 Nash, W.A .............. 326 Sakata, T ................ 336Kuak, Y.C ...... ........ 244 Nayfeh, A.H ............. 242 Sanborn, D.M ............ 303Kukkola, T ............. 380 Nelson, H.D............ 422 Saravanja-Fabris, N ......... 370Kuratani, K....... . .438 Nelson, R C ............. 350 Sato, K ................ 315Laithier, 5 E .. .. .. ... 295 Nemat-Nasser, S ... ... ... 271 Sawamoto, M ............. 280Lapini, G ..... ........ 426 Ng, S.-L ................ 443 Scerbo, L.J ............ 314Laratta, A ..... ....... 433 Nguyen, XJT............ 253 Schmitz, F.H ............. 361Large, J.B. 318 Ni, R.H ................ 307 Schneider, A.J .... ....... 292Larsson, L ........ 407 Nordman, R ........... 430 Schroder, E ............. 285Laurenson, R.M .. ........ 418 Nowinski, J.L . . . ... . 301 Schuster, G.M ............ 287Lazzeri, L ........... 382,383 Nuske, D J ............ 373,374 Schwerdtfeger, H ..... .... 425Leblois, L.C..... ... . . 381 Nuttall, S.M .. .. ...... 411 Seehra, M.S ............. 288Ledbetter, R H... ..... 355 O'Callaghan, M.J.A . ....... 326 Segal, D.J ............... 399Lee, E H...... ..... .. 293 O'Leary, T.R .... ...... 265 Setiawan, B ............. 333Lees, A.W . .. . . 445 Obi, C .. 237,238 Shafer, B.P ............. 296Lehmann, E.J ........ 259 Ochiai, Y..... ... 386 Shaffer, G.M ............ 402Lennox,W.C ....... 244 Ohkami,Y .. ..... 437 Shah, M.P ......... .. 392,393Likins, P.. .437 Ohmata, K ......... .342 ShimovetzR.M .......... 389Lin, J ........... 247 lkah-Avae, B E ........... 405 Shirota, K ............... 438Lindgren, B.J.......... 371 Olson, D A ..... ... .... 390 Sidar, M ................ 349Lindsey, L.G .. .. 434 Owzar, A ... 403 Siorek, R W .............. 348Liu, Y.K ............ 363 Paidoussis, M.P. .... 295 Simmons, P E ........... 444Longinow, A. 357 Parkinson, A G.. ...... 419 Skaistis, S.J . .,. ....... 344Lu, Y.P .... .340 Pattabiraman, J. ...... 328 Skjeltorp, A ......... 266,267,Lund, J & ... 412 Patterson,W.N....... 441 268,269,270Lutton, R J. 360 Penzien, J ...... 299 Smith, C.S .337McCarthy, M F. .265 Pestorius, F.M ... 345 Smith, D.L ......... 389McWhannell, D C 248 Pierce, G A. 312 Smith, J B ... ....... 364Ma, S M 379 Pollmann, E....... 425 Smith, J.D ........... 304Maeiawa, S 276 Possa, G .... 426 Soedel,W ............. 332Maglioizi, B. 352,353,354 Privitzer. E 357 Soong, T T ...... ..... .358Mallett, R L 293 Quinn, B E 397 Soroka, W.W..........241

93

Sorsdal, S...............289 Toda, Y. .............. 438 Wellstead,PRE ......... 373,374Steers, L. L.............. 398 Troppens,l D............. 317 Wiland, J.H .............. 366Stewart, J...............319 Tsujimoto, Y.............311 Wilkinson,T.L ............ 316Stewart, R.M ............ 369 Ungar, A ................ 278 Williams, R.S ............. 274Stinchcomb, W.W .......... 274 Urlichs, K .... .......... 446 Windett, G.P ...... ... 384, 385Stockdale, WXK........... 442 Utz, W. R................246 Winer, W.0Q.............. 303Strona, P.P ... ........... 383 Van Atta, F.A ............ 372 Wong, C ................ 437Suggs, C.W .............. 364 Vigran, T.E .............. 289 Woo,JL. ............... 302Sundararajan, C...........313 Wachel, J.C .............. 321 Woodford, DAi............416Suzuki, T............... 386 Walter,W.............. 415 Woolf, A................294Takasu, S. .............. 280 Warwick, J. E.............435 Worsfold, J.H ............. 445Takatsu, N .............. 315 Watanabe, T ............. 276 Wu, J.J.................298Taylor, R.L.............. 249 Watson, E.E ............. 291 Yang, J.Y ............... 331Thomas, C.B............. 415 Weiss, R................ 235 Yang, T.Y ............... 329TobeT.... ............ 315 Wel ford, G. D............ 362 Young, AM .............. 286Toda, H ............... 272 Wellford, L.C., Jr ...... ... 334

94

CALENDAR

MEETING DATE LOCATION CONTACT

1977MAR

NOISEXPO '77 14-17 Chicago, IL NOISEXPO '77, 27101 E. Ovlett Rd,Say Village, OH 44140Tale. (216) 835-0101

Structures, Structural Dynamics and 21-23 San Diego, CA ASME or AIAA Hq.Materials Conference, AIAA

15th Midwestern Mechanics Conference 23-25 Chicago, I L Prof. T.C.T. Ting, Dept. of MaterialsEngrg., Univ. of Illinois at ChicagoCircle, Box 4348, Chicago, IL 60680

Structural Dynamics Specialists Conference 24-25 San Diego, CA AIAA Hq.

Gas Turbine Conference and Products Show, ASME 27-31 Philadelphia, PA ASME Hq.

Joint Railroad Conference, IEEE/ASME 30-2 Washington, D.C. IEEE Hq.

APR

American Power Conference, Ill. Inst. Tech. 18-20 Chicago, IL R.A. Budenholzer, Dir.APC c/o lIT, 10W. 35th St.Chicago, IL 60616

Design Engineering Conference and Show, ASME 18-21 New York, NY ASME Hq.

Mini-Conference on Transportation 19-21 Ann Arbor, MI Highway Safety Research Institute,The University of MichiganAnn Arbor, M1 48109Tale. (313) 764-2168

Diesel and Gee Engine Power Conference 24-281 Dallas, TX ASME Hq.and Exhibit, ASME

IES Annual Meeting 24-27 Los Angeles, CA IES Hq.

9TH Space Simulation Conference 26-28 Los Angeles, CA IES Hq.IES-AIAA-ASTM-NASA

International Conference - Tribology April Cambridge, MA Lt. R.S. Miller, Code 211Office of Naval Research, Ballston

Tower No. 1, Arlington, VA 22117Tale. 692-4421

MAY

23rd International Instrumentation Symposium 1.5 Las Vegas, NV ISA Hq.

31st Annual Technical Conference, ASQC 16-18 Philadelphia, PA R.W. Shearman, ASOC Hq.

93rd Meeting of the Acoustical Society of America 17-20 State College, PA J.C. Johnson, Chairman, ASA

95

CALMNAR

MEETING DATE LOCATION CONTACT

JUNE

Fuels and Lubricants Meeting SAE 7-9 Tulsa, OK SAE Hq.

Applied Mechanics Conference, ASME 14-16 New Haven, CT ASME Hq.

Lubrication Symposium, ASME June St. Louis. MO ASME Hq.

SEPT

Vibrations Conference, ASME 26.281 Ch icaglo. I L ASME Hq.

96

CALENDAR ACRONYM DEFINITIONS AND ADDRESSES OF SOCIETY HEADQUARTERS

AFIPS: American Federation of Information CCCCAM: Chairman, c/o Dept. ME, Univ. Toronto,Processing Societies Toronto 5, Ontario, Canada210 Summit Ave., Montvale, N.J. 07645

IEEE: institute of Electrical and Electronics EngineersAGMA: American Gear Manufacturers Association 345 E. 47th St.

1330 Mass. Ave., N.W. New York, N.Y. 10017Washington, D.C.

IES: Institute Environmental SciencesAIAA: American Institute of Aeronautics and 940 E. Northwest Highway

Astronautics, 1290 Sixth Ave. Mt. Prospect, III. 60056New York, N.Y. 10019

IFThMM: International Federation for Theory ofAIChE: American Institute of Chemical Engineers Machines and Mechanisms, US Council for

345 E. 47th St. TMM, c/o Univ. Mass., Dept. ME, Amherst,Now York, N.Y. 10017 Mass. 01002

AREA: American Railway Engineering Association INCE: Institute of Noise Control Engineering59 E. Van Buren St. P.O. Box 3206, Arlington Branch,Chicago, III. 60605 Poughkeepsie, N.Y, 12603

AHS: American Helicopter Society ISA. Instrument Society of America30 E. 42nd St. 400 Stanwix St.. Pittsburgh, Pa. 15222New York, N.Y. 10017

ONR: Office of Naval ResearchARPA: Advanced Research Projects Agency Code 40084, Depx. Navy, Arlington, Va. 22217

ASA: Acoustical Society of America SAE: Society of Automotive Engineers335 E. 45th St. 400 Commonwealth DriveNew York, N.Y. '0017 Warrendale, Pa. 15096

ASCE: American Society of Civil Engineers SEE: Society of Environmental Engineers345 E. 45th St. 6 Conduit St.New York, N.Y. 10017 London WI R 9TG, England

ASME. Americaa Society of Mechanical Engineers SESA Society for Experimental Stress Analysis345 E. 47th St. 21 Bridge Sq.New York, N.Y. 10017 Westport, Conn. 06880

ASNT American Society for Nondestructive Testing SNAME Society of Naval Architects and Marine914 Chicago Ave. Engineers. 74 Trinity PI.Evanston, II. 60202 New York, N.Y. 10006

ASOC- American Society for Quality Control SVIC. Shock and Vibration Information Center161 W. Wisconsin Ave Naval Research Lab., Code 8404Milwaukee. Wis 53203 Washington, D.C. 20375

ASTM American Society for Testing and Materials URSI-USNC International Union of Radio Science - US1916 Race St National Committee c/o MIT Lincoln Lab.,Philadelphia, Pa 19103 Lexington, Mass. 02173

DEPARTMENT OF THE NAVY,NAAL RESEARCH-, ABQRATORYj, CODE 8404, POTG NDFEAID

SHOCK ~ ~ 44 -664 VIRTO ENOMO C-NTEWashngth~pC.~d375 -DEPAkTMENfT ,OF THE ,NAVY

OFFICIAL 6U6A&6Do3~PENALTY- FOR PRIVATE USE, $300.

THE SHOCK AND VIBRATION DIGEST

VOLUME 9 No. 2 Febm~ary 1977

EDITORIAL 19 JiINITE...ELEMENT P9DELING!67 NATUIFAL YJBRAWt ON

1 Director Notes PROALENIS,2 Editors Rattle Spac gV rsna Murty

ARTICLES AND REVIEWS 38 Book Reviews)

3 Feature Article (, BSORBERs AD CURRENT NEWSISOLATORS 0 TORSIONAL'VIBRATION; 40 News BriefsJ.M. Vance 41 Short Courses

7 Literature Review I BST-AT F~ROM AE CURRENT/ &LITERAVURE.9 IYNAMIC $TIFFNESS AND

DANIPINGO'FIBER.REN1 43 Abstract Contents

'YORCED CQNPOSINTE.ATEJ 44 Abstracts: 77.232 to 77-447RIALS; 92 Author IndexR.F. Gibson and R. Plunkett

CALENDAR