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Accordingly, it is noted that the same behavior for the lower mode, namely, the hump-backed curves and the phase Viscous Damping in Flexural approaching c„ for large values of Kpi, is obtained if, of Bars' instead of putting Kelvin damping in the extensional law, one were to put Maxwell-type damping in the shear law. In this J. M. McCORMICK, Jr.2 One of the curves given by the case, then, the behavior of the upper mode would also be ac- author in his Fig. 2 (that for the upper mode for large damping) ceptable. The author's approach, Laplace transform, would appears to be incorrect. The author shows the phase velocity, probably be unmanageable for Maxwell-type damping and the

c2, decreasing monotonically as the reciprocal of the wave length, objection would again have to be made concerning the necessity Downloaded from http://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/27/2/361/5443553/361_1.pdf by guest on 26 September 2021 Kpi, increases. Further, the author asserts in the text that "if of viscoelastic compressibility in the three-dimensional laws the range of Kpt2 is extended indefinitely, c2 tends to zero pro- if the Maxwell-type shear relation and an elastic extensional ducing short-wave-length cutoffs." It would seem, law are to be consistent. To assume that both laws have however, that while C2 initially decreases with increasing Kfii, standard-solid type behavior seems to be the only simple, physi- it eventually reaches a minimum and then increases monotoni- cally meaningful representation for the . Again cally and without a bound as Kincreases indefinitely. the Laplace-transform approach would probably be unmanage- That this is so may be seen by a rather simple calculation able mathematically for this type of law. [putting a wave form with real frequency and complex wave number into the governing differential equation, the author's Author's Closure Equation (7), and solving for the ratio of the frequency to the real part of the wave number as a function of the frequency], The solution of the beam equation presented in this paper is but it is perhaps more illuminating to reason qualitatively from written for specified initial and boundary conditions, the excita- consideration of the way in which the viscoelasticity is brought tion chosen being that of a half-period sine wave impact. The into the problem. response consists of the sum of the transient vibrations and the The author has assumed the extensional law to be of the Kelvin forced vibrations. type while retaining the elastic law for the shear components. The emphasis in the analysis is on the free vibrations since they (These two laws are consistent with each other only if they are represent the really significant part of the boundary-value prob- derivable from three-dimensional stress-strain laws, which lem. must then exhibit viscoelastic compressibility, whereas most The object of the dispersion curves shown in Fig. 2, including recent in viscoelasticity seems to favor using viscoelastic the one for large damping, is to exhibit the effect of damping on deviatorie relations with elastic compressibility.) It is thus to the free vibrations of the system. This can be done effectively by be expected that the viscoelastic effects will be evident only plotting as a continuous curve the wave velocity of free vibra- where the extensional law substantially influences the behavior. tions as function of a real wave number. The wave velocity of The author's damping curves (Fig. 3) tend to confirm this view. free damped vibrations is simply defined as the product of the frequency of damped free vibrations and the corresponding wave As the author points out in his introduction, the damping in length as given in Equations (38) and (39) of the paper. both modes increases at first with increasing Kp.2, but as the lower mode becomes more and more a pure shear mode, the Damped frequencies of free vibrations are the imaginary parts damping decreases and becomes negligible. The upper mode, of the roots zi.2 and z3., of the quartic equation (366). These however, continues to be influenced strongly by the extensional roots are defined in Equation (37a), where Z1.2 corresponds to the law and the damping continues to increase with increasing lower mode of wave and Z3.4 to the upper mode of wave KfXi. Qualitative predictions can also be made concerning the motion. dispersion curves (Fig. 2). The lower mode becomes essentially These so-called frequencies of damped vibrations are not the a shear mode for large values of Kpz and the phase velocity natural frequencies but form, when plotted as a continuous accordingly approaches the velocity of shear waves in an infinite curve, the locus for the discrete frequency spectrum characteristic medium. The upper mode, on the other hand, approaches a of a particular beam. The characteristic values are governed velocity [(E/p)l/t in the elastic case] which depends on the by the boundary conditions and determined as explained in the complex modulus, a quantity proportional to the real part of paper. the wave number for Kelvin-type damping. Thus the phase The discussion and analysis of the quartic equation (366) for velocity for this mode, with Kelvin-type clamping, must increase real values of K\u2 are therefore essentially an analysis of the without limit if Kpi is taken large enough, as long as the viscosity effects of damping on the frequency response in free vibrations of is nonzero. The author's curves for the upper mode in Fig. 2 the system. The terminology used when referring to "short- do not indicate behavior of this type; his low damping curve is wave-length vibration cutoffs," or to "frequency cutoffs" not extended far enough to tell and his high-damping curve means of course "cutoffs of free vibrations." exhibits a quite different behavior. The object of Fig. 2 and of Fig. 3 thus is to show that, no matter how small the damping is, as long as pi > 0, the upper The author's choice of stress-strain laws was apparently mode Ca will eventually reach critical damping as the wave length motivated by a desire to obtain damping curves which agree is diminished indefinitely. This means, of course, that wave with experimental results, like those for the lower mode in Fig. 3. of free vibrations, which are the object of this discus- 1 By M. K. Newman, published in the September, 1959, issue of the sion, tend to zero as wave lengths are diminished indefinitely. JOURNAL OF APPLIED MECHANICS, vol. 26, TRANS. ASME, vol. 81, It is also to be noted that frequency cutoffs may occur in the series E, pp. 367-376. • Institute of Flight Structures, Department of Civil case of the wave velocities ci of the lower mode of wave motion, and Engineering Mechanics, Columbia University, New York, N. Y. provided pi M,. In that case, there will exist a range of wave

Journal of Applied Mechanics june 1 9 6 0 / 361

Copyright © 1960 by ASME discussion numbers K/J.2 for which the damping factor ft ^ 1. In other Dissipation occurs in the region of the frequency spectrum where words, the humpbacked ft-curve in Fig. 3 may cross the value periods of vibrations are comparable with the of relaxation ft = 1 twice. For the range of values of Kfx2 where ft ^ 1, no of extensional strain. free vibrations exist. It was, however, pointed out in the paper The governing differential equation (7) of the paper, with that the values of the viscosity coefficients f corresponding to damping taken zero, is known to be a good approximation to values of pi < Mi cover a very large range of values that can be flexural wave propagation up to the very high frequencies. made to satisfy every conceivably practical condition of damping. The relative content of extensional and shearing strain trans- All this is in no way in contradiction to the argument advanced mitted by the lower mode of wave motion depends on the wave by the discusser. A wave form with real frequency and complex length or frequency. wave number presupposes a forced excitation of the system, In the range of frequencies starting at zero, and where the When substituted into the governing differential equation (7), wave velocities exhibit considerable dispersion, the strains wave velocities will be obtained that tend to infinity as the real propagated are predominantly extensional. The shearing strains part of the wave number is allowed to increase indefinitely. The that are transmitted in this range of frequencies are of the order effect of damping, however, is, in this case, completely obscured of small corrections. Consequently, neglecting viscoelastic

because the forced frequency response of the viscoelastic system damping in shear in this range can have only small effects even Downloaded from http://asmedigitalcollection.asme.org/appliedmechanics/article-pdf/27/2/361/5443553/361_1.pdf by guest on 26 September 2021 is insensitive to damping, or, in other words, the viscoelastic dis- though the coefficient of shear viscosity is not zero. persion curve in forced excitation is the same for all values of An approximation to thermoelastic damping in flexure can damping. therefore be obtained by relying entirely on extensional visco- The same result can also be obtained by substituting p = ioi provided the relaxation period of the strain is properly into Equation (20a) of the paper and allowing a> to become very chosen to fall within the range of frequencies where the exten- large. The complex wave number will then be of the form sional strain is the predominant component of the wave.

_ 1 + i Q}'/* Beyond this range, as wave lengths diminish still further, this mode tends to become more and more a pure shear mode. ~ V2 clfi>/> Thermoelastic damping in pure shear is zero when steady vibra- The wave velocity C2 which is the ratio of the frequency to the tions are applied. real part of the wave number is then seen to tend to infinity as Very few solids behave in fact, even approximately, like the wave lengths are diminished indefinitely. This can also be seen Voigt or the Maxwell model. For this reason, it has been sug- by noting that the wave velocity c% of the upper mode depends gested that the dynamic behavior of materials might be approxi- on the complex modulus, a quantity proportional to the real part mated more adequately in terms of a spectrum of relaxation of the wave number for Voigt or Kelvin-type damping. rather than by a single relaxation time. The choice of extensional viscoelasticity of the Voigt type to- In the simpler model described in this paper, advantage is gether with elastic shear components of stress is an approxima- taken of the dispersion characteristics in the lower mode of tion made necessary by the fact that viscoelastic damping in wave motion of the Timoshenko beam equation in conjunction shear introduces into the partial differential , with a properly chosen single relaxation of extensional strain. Equations (5) and (6) of the paper, coupling terms which make The governing differential equation (7) may therefore be con- it impossible to eliminate the rotational co-ordinate i/' and thus sidered, in the absence of the coefficient of shear viscosity, as an to combine Equations (5) and (6) into one Equation (7). approximation to thermoelastic damping in flexure which applies In viscoelasticity of the Voigt-Kelvin type, the coefficients in the region of the spectrum where propagation of shearing of viscosity are derived from the three-dimensional stress-strain strain is negligible. laws as coefficients of first-order approximations in the time An estimate of the range of validity of this analysis can be rates of strain. These coefficients of viscosity cannot be ex- established by a comparison of the clamping curves, obtained for pressed in a consistent manner as is the case with the elastic various materials using extensional relaxation, with the results constants for a homogeneous, isotropic, and perfectly elastic obtained from thermoelasticity in flexure. solid. This is inherent in the assumption of viscoelasticity. Nevertheless, viscoelasticity can be looked upon as a reasona- ble approximation to the processes of dissipation of mechanical . On Supersonic Wind Tunnels With For example, in a specific case, viscous energy dissipation in 1 shear may be negligible when compared to energy dissipation Low Free-Stream Disturbances in normal strain. This may happen over a whole band of fre- quencies where some particular phenomenon such as the thermo- THOMAS VREBALOVICH.2 The author describes clearly the elastic effect is to be approximated and where transmission of problem of trying to reduce and determine the causes of free- shearing strain is insignificant. stream disturbances in supersonic wind tunnels. He has much In addition, it may happen that the phenomenon to be approxi- experimental evidence to indicate some of the sources of these dis- mated does not extend over that part of the frequency spectrum turbances, and describes the manner in which many of these where the shearing strains transmitted are the predominant part disturbance sources may be eliminated, or at least minimized. of the wave motion. Some of the results of a series of experiments conducted at the The aim in this problem is to devise a viscoelastic model of Jet Propulsion Laboratory3 in the 12-in. and 20-in. supersonic thermoelastic damping in flexural vibrations of reeds. In the case of thermoelasticity, the arises from tem- 1 By M. V. Morkovin, published in the September, 1959, issue of perature relaxation across the reed. Irreversible conversion the JOURNAL OF APPLIED MECHANICS, vol. 26, TRANS. ASME, vol. 81, series E, pp. 319-324. of mechanical energy into heat appears as internal friction in 2 Jet Propulsion Laboratory, California Institute of Technology, the range of frequencies where periods of vibrations are com- Pasadena, Calif. parable with the relaxation time for conduction of heat across a 3 This paper presents the results of one phase of research carried specimen. out at the Jet Propulsion Laboratory, California Institute of Tech- nology, under joint sponsorship of the Department of the Army, In the viscoelastic model, a similar mechanism is used. Fric- Ordnance Corps (under Contract No. DA-04-459-Ord 18), and the tion arises from strain relaxation in extensional viscoelasticity. Department of the Air .

362 / j u n e 19 6 0 Transactions of the ASME