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Temperature-frequency dependence of dynamic properties of damping materials
Journal o f Stored and Vibration (1974) 33(4), 451---470
TEMPERATURE-FREQUENCY DEPENDENCE OF DYNAMIC PROPERTIES OF DAMPING MATERIALS D. I. G. JONES Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio 45433, U.S.A. (Received 31 August 1973, and hz revised form 5 November 1973)
Some of the simpler and more reliable techniques for measuring the complex modulus properties of elastomeric and other materials, as functions of frequency, temperature and strain, are reviewed. Use is made of the well established "temperature-frequency superposition" principle to reduce the data to a standardized form useful for engineering purposes. Specific examples of the techniques for reducing the data are discussed. Possible future application of the results to automated data processing and reduction is briefly discussed.
1. INTRODUCTION Ferry [1] and others [2--4] have shown that, provided certain assumptions are made, the dynamic mechanical properties of certain types of "thermo-rheologieally simple" materials, such as many of the elastomers, in the temperature-frequency domain, depend on a single compound variable which combines the effect of both frequency and temperature. In view of the fact that all techniques for measuring the dynamic mechanical properties of materials cannot obtain totally reliable data over a broad frequency range at all relevant temperatures, this "temperature-frequency equivalence" principle is extremely valuable for extrapolating data into regions where direct measurement is difficult or impossible. At the present time, a great variety of new or modified measurement techniques and instrumentation are emerging, along with many new damping materials. However, the need to exercise great caution in measuring the properties of materials is as great as ever and has been neglected in many recent investigations. Therefore, the need for more widespread application of the temperature-frequency equivalence principle for producing reliable data for engineering use is becoming imperative. Otherwise, potential users of damping materials can expect to be confused by a flood of incomplete and unreliable data on innumerable commercial and experimental materials in the future. It is the purpose of this paper to review several relatively simple and inexpensive measurement techniques which can be used to obtain reliable complex modulus data for a wide variety of damping materials, and how the "temperature-frequency equivalence" principle can be used to reduce the data in a reliable and efficient manner, even for materials which are not normally regarded as being thermo-rheologically simple. The materials evaluated include a broad temperature range filled silicone elastomer, a high damping acrylic adhesive, a twopart RTV silicone potting compound, a filled nitrile rubber, a blend of several polymers and a non-elastomeric vitreous enamel for very-high-temperature damping. These materials are typical of the kinds of materials available for providing damping in the temperature range -100~ (-72~ to 2000~ (1128~ albeit with some notable gaps! It will be shown that, even though these materials are not normally considered to be simple, the temperaturefrequency equivalence principle nevertheless allows a mass of scattered data of otherwise 451
D . I. G . J O N E S
ited usefulness to be gathered together and used to fully define the complex modulus pperties of each material over an enormous frequency range. The resulting curves of reduced dulus and loss factor versus frequency and temperature, over a very broad range, are of at value because they allow experimenters to correlate test results with materials data ter almost any conditions, allow designers far more flexibility than they would otherwise ,e with limited data, and allow potential users to evaluate the relative merits of many terials under a variety of operating conditions. This approach to data reduction a n d sentation should be capable of application, with great advantage, to a wide variety of terials besides those tested here.
2. CHARACTERIZATION OF MATERIAL BEHAVIOR .'onsider a sample of a damping material under harmonic strain excitation. If the material near, the relationship between stress and strain may be expressed as a = ED(CO,T)[I + i~/n(co,7")] e
(1)
:re ED(co,T) is the real part of the complex Young's modulus, for extensional strain, and ~, T) is the loss factor, which is a partial measure of the damping capability of the material st of nomenclature is given in the Appendix). In general ED and tlD are functions of the Luency, co, and the temperature, T. The essential simplicity of the temperature-frequency ivalence principle [1, 2] lies in the assertion that the values of ED and r/D at different peratures and frequencies obey the very simple relationships
Eo(co, T) = (Tp/To Po) ED(co:zr),
(2)
r/o(co, T) -- r/o(co~r),
(3)
re C~ris a function of temperature known as the shift factor and To is a reference temperaon the absolute scale. Equations (2) and (3), if valid, imply that if c0t is suitably chosen for I temperature, the two variables ~o and Thave been reduced to a single variable co~r, and :hree-dimensional relationship between Eo, co and Tis reduced to a simple two-dimensional tionship, and similarly for 11o. This enormous simplification can, at a stroke, reduce ".ns of graphs o f Eo and !ID versus co, for many temperatures, and often for only a few test Its and a narrow frequency range, to a single pair of curves with many points. These same fiderations apply to the complex shear modulus, Go(1 + bl~). he essence of the whole procedure, however, lies in the selection ofc~r for each temperature. is best done empirically, at least initially, by selecting the reference temperature To (in "ees absolute) arbitrarily but sensibly somewhere near the middle of the range of temperas for which useful data are available. This selection is initially a matter of convenience luse a more rational selection can always be made later when one is more familiar with the tvior of the material. For temperature To, one can then define ~Zr = 1 and so plot En and irectly against co~r for the points available. The aim for the next adjacent temperature is ;lect c~r in such a way that the graphs o f ( T p / T o P o ) E n and !1o versus coc~r match as closely ,ossible, in position and slope, the points already drawn. Several selections of c~r may .~to be made before the results appear satisfactory. The process is repeated step by step ill temperatures available. The resulting curves of (Tp/ToPo)ED and tin versus ccr should mooth and have minimum scatter but, since human judgment is involved, some differ:s between results obtained by different individuals will be noted. This variability can be iced by comparing the results for many materials and verifying that the curve ofccr versus '.emperature difference ( T - To) is universal for all intents and purposes provided that To
453
DYNAMIC PROPERTIES OF DAMPING MATERIALS
is suitably chosen for each material. This is usually accomplished on a trial and error basis. The values of To were determined for the materials investigated here in such a way as to ensure that the Ctr versus (T-- To) curves for each material collapse on the same graph. It is important to realize that this is an experimental procedure which can be made to work for any material having properties which correspond to some part of the ~r versus (T-- To)" curve. This is true even for the enamel tested in this investigation or for any other which behaves in the same general way. It will not be true for materials exhibiting markedly different behavior, but this can only be determined for each material tested on its own merits.
3. MEASUREMENT TECHNIQUES A great variety of test techniques have been developed for measuring the complex modulus properties of materials [5-11]. Some of the techniques are difficult to use and require complex electronic instrumentation. The techniques used to obtain the data evaluated in this paper are relatively simple and include the classical resonance technique [9] and the vibrating cantilever beam technique with the damping material applied as a single external coating [5, 11], a symmetric external coating [l 1] or the inner layer of a three layer sandwich beam [12]. 3.1. RESONANCETESTS The types of specimen'used in the resonance tests are illustrated in Figure 1. A typical test setup block diagram is shown in Figure 2. In the tests, the specimens are excited harmonically with constant acceleration level at the shaker table, and the response of the mass, M, is measured by means of an accelerometer. The input acceleration, J(, the output acceleration, 7, the resonance frequency,f,, and the temperature, T, are measured for several values of the mass, M. From these measured quantities, the real part of the Young's modulus, Eo, and the loss factor, tl~, are derived [9, 10] from the-relationships r/o = (A 2 " I) -''2.
(4)
Eo = 4r?f ~ h o ( M + m]3)/S(1 + flSZ[S2).
(5)
Alternatively, for the shear specimen [9], Go = 4n2f,2 hD(M + m]3)[1 + (h~/6RZ)][S.
(6)
By varying the input level, the material properties can be measured over a wide range of peak strain levels but, in the present tests, levels were kept low enough to remain within the range of linear behavior. The primary advantages of the resonance test are simplicity, both in
~ Tension - c~r specimen
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Figure 1. Resonance test specimen [C, control accelerometer, P, pickup accelerometer].
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D. I. G. JONES
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testing and analysis, and the capability of assessing the effects of strain amplitude. Disadvantages are the necessity ofchanging mass to vary frequency and the difficultyof changing frequency and pre-load independently. The use of several specimens of varying geometry is a useful way of ensuring that data obtained are reliable and cover the appropriate range of parameters. Specimens should be simple and not have too low a height to diameter ratio to avoid errors resulting from highly non-uniform strains in the specimen. At very high frequencies, care should be taken to avoid standing waves in the specimens, which will render the above equations invalid. 3.2.
VIBRATING BEAM TESTS
The types of specimen used in the vibrating beam tests are illustrated in Figure 3. A typical block diagram is shown in Figure 4. In the tests, the beams were generally excited by means of a magnetic transducer at the free end and the response picked up by another magnetic transducer near the root. By using the graphic level recorder, response spectra comprising graphs of transverse peak velocity versus frequency were plotted and the resonant frequencies, f,, and half-power bandwidth, Aft, noted for many temperatures. For the high-temperature tests, however, for which suitable transducers were not available, the beam was instead excited by a shaker through an impedance head and a pillar made of Hastelloy-X [13]. In this case, the force transmitted back to the transducer from the beam was measured, with the force due to the pillar inertia cancelled electronically, and the same quantities, Jr, and Aft, measured. For reliable results with commercial versions of this method, it is usually necessary to use beams made of a relatively low modulus material, such as aluminum, with machined roots as indicated in Figure 3 for reduced root damping, not more than about 0.05 inches thick nor shorter than about 7 inches. However, for the enamels it was necessary to use another material because of the high temperatures involved and this made testing more difficult. The advantage of the method is that it is reasonably simple to use, errors can be assessed and kept within limits [12] and a wide band of frequencies and temperatures can be covered with a single specimen. The disadvantage is that only low-strain-level data are obtainable, within
DYNAMIC PROPERTIES OF DAMPING MATERIALS
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456
D.I.G. JONES
the linear range for the damping material. This, however, is not usually a problem since one would use another method if such data were needed. 3.2.1. "Oberst beam" tests For the beam with the damping material coated on one side only, the complex Young's modulus is derived from formulae [5, 11 ] due originally to Oberst. These are: 1 + 2en(2 + 3n + 2n 2) + e 2 n 4 z ~ =
,
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rl.._L= en [3+6n+4n2+2enS+ean" ~, r/D (1 + en) \ i ~ne'~Z'-+"3~n~3-+---~n 4 )
(7)
(8)
where Z 2 = (I + pDn/p)(f,[fo,) z, e = E r I E and n = h d h . in these formulae, Z 2 is calculated from the measured resonance frequency, f., of the nth mode of the damped beam and the measured frequency, fo,, of the undamped beam and e is then deduced from equation (7). t/o is then calculated from equation (8), this value of e and the measured value of t/, = Af,[f, being used. These equations give reasonably accurate results provided that Z 2 - 1 >10.1 or so [5, 11].
3.2.2. Modified "Oberst beam" tests For the beam with the damping material coated symmetrically on b0th-sides, the complex Young's modulus is derived [11] from the formulae Eo = E ( Z 2 - 1)/[8n 3 + 12n 2 + 6n], ,7o = ~ , Z ~ l ( Z
~ -
1),
(9) (10)
where Z 2 = (I + 2pon]p)(fdfo,) 2 and e and n are defined as before. In these formulae Z 2 and qs are determined as before. Again, the equation gives reasonably accurate results whenever Z 2 - I >i 0-1 or so [5, 11]. These formulae are also applicable directlyto the hightemperature specimens provided that the beam is coated symmetrically. A similar technique is also applicable for beams with multiple constrained layers added [14].
3.2.3. Sandwich beam tests Finally, for the symmetric sandwich beam, the approach is based on a set of equations due to Kerwin and others [15], reduced by the author and Nashif [12] to an uncoupled form. The shear modulus, GD, and loss factor, r/o, of the viscoelastic material may then be derived from the equations Go =
[(A - B) -- 2(a -- B) ~ - 2(Aq,) 21[~hh~, ~,~IL21 (1 -- 2A + 2B) 2 + 4(dq,) 2 '
(11)
rl'o = a q d [ A - B - 2(A - B) 2 - 2(Aq,)2],
(12)
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(13)
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(14)
where
and
DYNAMIC PROPERTIES OF DAMPING MATERIALS
457
For most elastomeric materials in the rubbery and transition regions, Eo ~ 3Go and
4. TEST RESULTS 4.1. FILLEDSILICONEELASTOMER[16] This material was evaluated by the resonance technique, square cross-section cylindrical specimens being used, with b = 0.50 in (12.7 mm) and ho = 0.86 in (21.8 mm) according to Figure I. Some of the measured values of ED and r/o for several temperatures and frequencies are shown in Table 1. To was taken arbitrarily as 45~ (505~ (281~ Graphs of ~r versus temperature are shown in Figure 5. Graphs of (To/T)ED and r/D versusf~r are plotted in Figure 6. The values of err were selected to give the best empirical fit of this data. It is seen that even the data at -100~ (200~ did not go far into the transition region, which is beyond l0 s Hz at 45~ (281~ as Figure 6 shows. The specimen link mass was 4.7 grams and the density was 0.048 lb/in 3 (1329 kg/m a) [17]. The data given here are limited to the linear region (max. strain ~< 10-3 in/in) for this particular material. Tests at higher strain levels were investigated also and showed non-linear amplitude dependent behavior. The results show that the data for many temperatures and frequencies can be collapsed on to a single curve and that the data obtained are sufficiently accurate and consistent for engineering purposes. 4.2. ACRYLICADHESIVE[18] This material was also evaluated by the resonance technique, square cross-section cylindrical specimens as illustrated in Figure 1 being used, with b = 0.525 and hD = 0.45 in. The mass of the link was 2.0 gm and the density of the material was 0.038 lb/in 3 (1052 kg/m3). Some measured test results for the tension--compression specimen are given in Table 2, for low input levels. The material is linear up to peak strain levels of well over I ~o, and so linearity was not an important factor in these particular tests. To was finally selected as 120~ (580~ 322~ so as to give a best fit to the experimental data and fit the previous value of~r. Graphs of(To/T)Eo and r/o versusf~r are plotted in Figure 7. Values ofct r are plotted versus temperature in Figure 5. It is seen from the results that Eo has a wide range of values and yet the scatter is reasonably small, even with r/o very large. The graphs of ~r versus temperature lie on the same curve as those for the filled silicone with this choice of To. 4.3. SILICONE POTTING COMPOUND 9
Again, the material properties were determined by a resonance technique, rectangular cross-section cylindrical specimens being used with bma~= 0"50 in (12-7 ram), bmi, = 0"45 in (11.4 mm) and hD = 1.00 in (25.4 mm). Measured results are given in Table 3. To was taken as 160~ (620~ 344~ Graphs of(To/T)Eo and 17D versusf~r are plotted in Figure 8. Values of a r are again plotted versus temperature in Figure 5, and again the curves of ~r versus temperature form a single curve. The link mass was 3.95 grams and the density was 0.038 lb/in 3 (1052 kg/m3). 4 . 4 . FILLED NITRILE RUBBER
[20]
This material was evaluated by the vibrating beam technique with the material coated symmetrically on both outer surfaces of the beam. The beam material was aluminum with a thickness of 0.032 inches and length L = 7 in. The damping material thickness was 0.I 19 in.
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4.5. POLYMERBLEND [21] The particular polymer blend tested here was a homogeneous milled mixture of ParacrilBJ, Paracril-D, Polyvinyl Acetate and Polystyrene, with the following specific formulation [20]: ParacriI-BJ Paracril-D Polyvinyl Acetate Polystyrene (Dylene 8) Zinc Oxide Anti-oxidant 2246 Dicumyl Peroxide SAF Black
50 P H R 100 50 100 10 1 7 50.
The specimens were cured at 280~ for 1 hour. Further details of specimen preparation are given in references [20] and [21]. The test specimens consisted of an aluminum beam 7 inches long, 0.5 inches wide and 0.032 inches thick, with a layer of the damping material 0.092 inches thick on each side for specimen A and 0.045 inches thick for specimen B. The density of the damping material was 0.041 lb]in 3 and that of the aluminum beam 0.100 lb]in s. The value of To was determined to be 230~ (690~ again on an empirical basis. Some of the reduced data for the tests on the two specimens is given in Table 5. Graphs of (To/T)Eo and ~/o versus foot are presented in Figure I0. Values of ~r as a function of temperature are plotted in Figure 5.
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The particular enamel tested [22] was applied as a slurzT to the two surfaces ofthe beam and fired at about I800~ The beam material was Hastelloy-X steel; which had a density of 0,286 Iblin 3 (7916 kglm3). Further coats were added as necessary and the beam re=fired until the thicknesses were uniform. Two beams were examined in these tests, although others were also tested [12], the first (A) being I,?5 inches (44.5 ram) long by 1.5 inches (38"I ram) wide by 0-030 inches (0.?6 ram) thick and having coating thickness ho = 0.0.185 inches (0.47 ram) and the other (B) 3.00 inches (76.2 ram) long by 0.5 inches 02"? ram) wide by frO? inches (I.78 ram) thick, and having a coating thickness ho = 0.0045 inches (0.11 ram). The density of the enamel was 0.09 Iblin 3 (2491 kglm3). Some of the observed data are recorded and reduced in Table 6. :Towas determined by trial and error to be 1250~ (1710~ 950~ Reduced data are plotted as graphs of (ToIT)ED and ~D versus'f0cr in Figure I I. Values of ccr are plotted v e r s u s temperature in Figure 5. )o7 : I =)l'"q
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5. UTILIZATION OF DATA The graph ofctr versus ( T - To), depicted in Figure 5, is very important because it represents a very general relationship which is accurate for a wide variety of materials. It can be shown, on the basis of a simple curve-fit, that ~r is given by Ctr -- [1 + ( T -- To)/2781-".
(15)
If, therefore, one is investigating the complex modulus data of any damping material and has test data in the form of blocks, each consisting of four numbers representing values of T, 9 f, Eo, and r/o, then one may semi-automate the process of reducing the data no matter how arbitrary the values and intervals of T a n d f m a y be. The general procedure which would have to be adopted is as follows: (1) make an initial guess for To; (2) compute Ctr from equation (15) with each T measured in degrees F at this point; (3) computefc~r and (To/T)Eo with To and T measured on the absolute scale at this point; (4) plot, by using a digital plotting routine, the graphs of (To/T)Eo and r/D versusfc~r; (5) from inspection of these graphs, select a new value of To and return to step (1); repeat until the best fitting data are obtained in step (4).
6. CONCLUSIONS
It has been shown in this paper that simple, well-established techniques are available for accurately measuring the complex modulus properties of damping materials. Provided that their limitations are recognized and respected, sufficient data can usually be obtained by use of each technique to completely characterize material damping behavior as a function of frequency and temperature over the range encountered in engineering applications. This is accomplished by conducting the tests over a wide range of temperatures and using the temperature-frequency equivalence principle to collapse the data to a single set of curves. The very general applicability of this principle has been verified for several dissimilar materials, ranging from a filled silicone elastomer through a polymer blend to a vitreous enamel, over a 550~ (300~ temperature range and a sixteen decade frequency range. This indicates that measurements obtained by using any reasonably accurate technique, even over a limited frequency range which would otherwise be of little value, can be collapsed for many temperatures to markedly extend the useful range of the measurement technique and thus obtain, on an engineering basis, data which have previously been obtainable only under more controlled laboratory conditions. Finally, the development of an approximate analytical relationship describing the variation of the shift factor with the temperature difference provides the basis for a semi-automatic process of reducing complex modulus data for damping materials no matter how arbitrarily the temperature and frequency values may be distributed, provided that the range is adequate. ACKNOWLEDGMENTS This work was sponsored by the Air Force Materials Laboratory, United States Air Force, under Project No. 7351, Task No. 735106. Thanks are due to A. D. Nashif, C. M. Cannon, R. W. Gordon and M. L. Parin for assistance with the tests, D. Gochoel and B. Dues for assistance with manuscript preparation and S. Askins for assistance with electronic instrumentation.
DYNAMICPROPERTIESOF DAMPINGMATERIALS
469
REFERENCES 1. J. D. FERRY 1961 Viscoelastic Properties of Polymers. New York: John Wiley and Sons, Inc. 2. J. D. FERRY, E. R. FITZGERALD,L. D. GRANDINE,JR. and M. L. WILLIAMS1952 Industrial and Engineering Chemistry44, 703-706. Temperature dependence of dynamic properties ofelastomers; relaxation distributions. 3. J. C. SNOWDON1968 Vibration and Shock hi Damped Mechanical Systems. New York: John Wiley and Sons, Inc. 4. B. J. LAZAN 1968 Damping of Materials and Members in Structural Mechanics. Oxford: Pergamon Press Ltd. 5. H. OBERST1952 Acustica (Akustische Beihefte) 4, 181-194. Ober die Dampfung Biegeschwingungen Dunner Bleche Durch fest Haftende Belage. 6. T. J. DtlDEK 1969 Journal of the Acoustical Society of America 46, 1384--1386. Damping material effectiveness measured by the Geiger-plate and composite-beam tests. 7. R.L. ADKINSJuly 1966ExperimentalAlechanics. Design considerations and analysis of a complex modulus apparatus. 8. J. L. EDWARDSand D. R. HICKS1972Journal ofthe Acoustical Society of America 52, 1053-1056. Useful range of a mechanical impedance technique for measurement of dynamic properties of materials. 9. D. I. G. JONESand M. L. PARtN 1972 Journal of Soundand Vibration 24, 201-210. Technique for measuring damping properties of thin viscoelastic layers. 10. C. M. CANNON,A. D. NASHIE,and D. I. G. JONES1968 Shock and Vibration Bulleth~ 38, 151-163. Damping measfirements on soft viscoelastic materials using a tuned damper technique. 11. A. D. NASHIF 1967 Shock and Vibration Bulletin 36, 37--47. A new method for determining damping properties of viscoelastic materials. 12. A. D. NASHIF1973 Shock and Vibration Bullethz 43, 145-151. Materials for vibration control in engineering. 13. A. D. NASHIF 1973 Proceedings of SEECO 73, Society of Electrical Engineers Conference, London. Damping of glass-like materials at high temperatures. 14. D. I. G. JONES,A. D. NASHIFand M. L. PARIN1973 Journal of Sound and Vibration 29, 423--434. Parametric study of multiple-layer damping treatments on beams. 15. D. Ross, E. E. UNGARand E. M. I~RWIN, JR. 1959 Structural Damping, American Society of Mechanical Engineers 49-87. Damping of plate vibrations by means of viscoelastic laminae. 16. BTR: a broad temperature range filled silicone elastomer supplied by Lord Mfg. Co., Erie, Pa. 17. 3M--467: an acrylic adhesive supplied by the Minnesota Mining and Manufacturing Company, Minneapolis, Minnesota. 18. SYI-~ARD188: a two part silicone RTV pottingcompound supplied by Dow Corning Corporation, Midland, Michigan. 19. PARACRm-BJ:a nitrile rubber formulation supplied by U.S. Rubber Company. 20. F. S. OWENS 1967 Shock and Vibration Bullethz 36, 25-35. Elastomers for damping over wide temperatures ranges. 21. F. S. OWENS1970 Air Force Materials Laboratory Report AFML-TR-70-242. Wide-temperature range free-layer damping materials. 22. VITREOUSENAMELCV 17214, manufactured by Chicago Vitreous Co., Chicago, Illinois. APPENDIX A b B D e E ED f fo. f. Go
NOMENCLATURE amplification factor I Y/-YI for resonance specimen; also a non-dimensional parameter breadth of cylindrical specimen non-dimensional parameter suffixdenoting damping material E~/E: the modulus ratio Young's modulus of beam material real part of complex Young's modulus of damping material frequency (Hertz) nth natural frequency of undamped beam nth resonant frequency of damped beam real part of complex shear modulus of damping material
470 h hD 1 L m M n
R S St T To X Z2 Ctr fl e r/D t/s tip ).n ~ p po
Po a co co~ a~o~
D . I . C . JONES thickness of beam thickness of damping material length of viscoelastic material in shear resonance specimen length of beam mass of viscoelastic material added mass on resonance test specimen hD/h: thickness ratio; also suffix indicating mode number radius of gyration of shear specimen about horizontal axis in Figure I (R = l/~/i2for rectangular specimen) load carrying area of damping material in resonance specimens non-load carrying (free) area of damping material in resonance specimens temperature reference temperature input acceleration for resonance specimen output acceleration for resonance specimen non-dimensional parameter temperature shift factor shape factor for cylindrical resonance test specimen; fl = 2.0 for unfilled clastomers and ~ 1-5 for filled elastomers strain shear loss factor loss factor of beam specimen extensional loss factor of damping material wavelength ofnth beam mode nth eigenvalue for beam density of beam; also density in general density of damping material where specific identification is necessary density at reference temperature stress circular frequency nth circular frequency of damped beam nth circular frequency of undamped beam
TEMPERATURE-FREQUENCY DEPENDENCE OF DYNAMIC PROPERTIES OF DAMPING MATERIALS D. I. G. JONES Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio 45433, U.S.A. (Received 31 August 1973, and hz revised form 5 November 1973)
Some of the simpler and more reliable techniques for measuring the complex modulus properties of elastomeric and other materials, as functions of frequency, temperature and strain, are reviewed. Use is made of the well established "temperature-frequency superposition" principle to reduce the data to a standardized form useful for engineering purposes. Specific examples of the techniques for reducing the data are discussed. Possible future application of the results to automated data processing and reduction is briefly discussed.
1. INTRODUCTION Ferry [1] and others [2--4] have shown that, provided certain assumptions are made, the dynamic mechanical properties of certain types of "thermo-rheologieally simple" materials, such as many of the elastomers, in the temperature-frequency domain, depend on a single compound variable which combines the effect of both frequency and temperature. In view of the fact that all techniques for measuring the dynamic mechanical properties of materials cannot obtain totally reliable data over a broad frequency range at all relevant temperatures, this "temperature-frequency equivalence" principle is extremely valuable for extrapolating data into regions where direct measurement is difficult or impossible. At the present time, a great variety of new or modified measurement techniques and instrumentation are emerging, along with many new damping materials. However, the need to exercise great caution in measuring the properties of materials is as great as ever and has been neglected in many recent investigations. Therefore, the need for more widespread application of the temperature-frequency equivalence principle for producing reliable data for engineering use is becoming imperative. Otherwise, potential users of damping materials can expect to be confused by a flood of incomplete and unreliable data on innumerable commercial and experimental materials in the future. It is the purpose of this paper to review several relatively simple and inexpensive measurement techniques which can be used to obtain reliable complex modulus data for a wide variety of damping materials, and how the "temperature-frequency equivalence" principle can be used to reduce the data in a reliable and efficient manner, even for materials which are not normally regarded as being thermo-rheologically simple. The materials evaluated include a broad temperature range filled silicone elastomer, a high damping acrylic adhesive, a twopart RTV silicone potting compound, a filled nitrile rubber, a blend of several polymers and a non-elastomeric vitreous enamel for very-high-temperature damping. These materials are typical of the kinds of materials available for providing damping in the temperature range -100~ (-72~ to 2000~ (1128~ albeit with some notable gaps! It will be shown that, even though these materials are not normally considered to be simple, the temperaturefrequency equivalence principle nevertheless allows a mass of scattered data of otherwise 451
D . I. G . J O N E S
ited usefulness to be gathered together and used to fully define the complex modulus pperties of each material over an enormous frequency range. The resulting curves of reduced dulus and loss factor versus frequency and temperature, over a very broad range, are of at value because they allow experimenters to correlate test results with materials data ter almost any conditions, allow designers far more flexibility than they would otherwise ,e with limited data, and allow potential users to evaluate the relative merits of many terials under a variety of operating conditions. This approach to data reduction a n d sentation should be capable of application, with great advantage, to a wide variety of terials besides those tested here.
2. CHARACTERIZATION OF MATERIAL BEHAVIOR .'onsider a sample of a damping material under harmonic strain excitation. If the material near, the relationship between stress and strain may be expressed as a = ED(CO,T)[I + i~/n(co,7")] e
(1)
:re ED(co,T) is the real part of the complex Young's modulus, for extensional strain, and ~, T) is the loss factor, which is a partial measure of the damping capability of the material st of nomenclature is given in the Appendix). In general ED and tlD are functions of the Luency, co, and the temperature, T. The essential simplicity of the temperature-frequency ivalence principle [1, 2] lies in the assertion that the values of ED and r/D at different peratures and frequencies obey the very simple relationships
Eo(co, T) = (Tp/To Po) ED(co:zr),
(2)
r/o(co, T) -- r/o(co~r),
(3)
re C~ris a function of temperature known as the shift factor and To is a reference temperaon the absolute scale. Equations (2) and (3), if valid, imply that if c0t is suitably chosen for I temperature, the two variables ~o and Thave been reduced to a single variable co~r, and :hree-dimensional relationship between Eo, co and Tis reduced to a simple two-dimensional tionship, and similarly for 11o. This enormous simplification can, at a stroke, reduce ".ns of graphs o f Eo and !ID versus co, for many temperatures, and often for only a few test Its and a narrow frequency range, to a single pair of curves with many points. These same fiderations apply to the complex shear modulus, Go(1 + bl~). he essence of the whole procedure, however, lies in the selection ofc~r for each temperature. is best done empirically, at least initially, by selecting the reference temperature To (in "ees absolute) arbitrarily but sensibly somewhere near the middle of the range of temperas for which useful data are available. This selection is initially a matter of convenience luse a more rational selection can always be made later when one is more familiar with the tvior of the material. For temperature To, one can then define ~Zr = 1 and so plot En and irectly against co~r for the points available. The aim for the next adjacent temperature is ;lect c~r in such a way that the graphs o f ( T p / T o P o ) E n and !1o versus coc~r match as closely ,ossible, in position and slope, the points already drawn. Several selections of c~r may .~to be made before the results appear satisfactory. The process is repeated step by step ill temperatures available. The resulting curves of (Tp/ToPo)ED and tin versus ccr should mooth and have minimum scatter but, since human judgment is involved, some differ:s between results obtained by different individuals will be noted. This variability can be iced by comparing the results for many materials and verifying that the curve ofccr versus '.emperature difference ( T - To) is universal for all intents and purposes provided that To
453
DYNAMIC PROPERTIES OF DAMPING MATERIALS
is suitably chosen for each material. This is usually accomplished on a trial and error basis. The values of To were determined for the materials investigated here in such a way as to ensure that the Ctr versus (T-- To) curves for each material collapse on the same graph. It is important to realize that this is an experimental procedure which can be made to work for any material having properties which correspond to some part of the ~r versus (T-- To)" curve. This is true even for the enamel tested in this investigation or for any other which behaves in the same general way. It will not be true for materials exhibiting markedly different behavior, but this can only be determined for each material tested on its own merits.
3. MEASUREMENT TECHNIQUES A great variety of test techniques have been developed for measuring the complex modulus properties of materials [5-11]. Some of the techniques are difficult to use and require complex electronic instrumentation. The techniques used to obtain the data evaluated in this paper are relatively simple and include the classical resonance technique [9] and the vibrating cantilever beam technique with the damping material applied as a single external coating [5, 11], a symmetric external coating [l 1] or the inner layer of a three layer sandwich beam [12]. 3.1. RESONANCETESTS The types of specimen'used in the resonance tests are illustrated in Figure 1. A typical test setup block diagram is shown in Figure 2. In the tests, the specimens are excited harmonically with constant acceleration level at the shaker table, and the response of the mass, M, is measured by means of an accelerometer. The input acceleration, J(, the output acceleration, 7, the resonance frequency,f,, and the temperature, T, are measured for several values of the mass, M. From these measured quantities, the real part of the Young's modulus, Eo, and the loss factor, tl~, are derived [9, 10] from the-relationships r/o = (A 2 " I) -''2.
(4)
Eo = 4r?f ~ h o ( M + m]3)/S(1 + flSZ[S2).
(5)
Alternatively, for the shear specimen [9], Go = 4n2f,2 hD(M + m]3)[1 + (h~/6RZ)][S.
(6)
By varying the input level, the material properties can be measured over a wide range of peak strain levels but, in the present tests, levels were kept low enough to remain within the range of linear behavior. The primary advantages of the resonance test are simplicity, both in
~ Tension - c~r specimen
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, Shear specimen
Figure 1. Resonance test specimen [C, control accelerometer, P, pickup accelerometer].
-
454
D. I. G. JONES
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testing and analysis, and the capability of assessing the effects of strain amplitude. Disadvantages are the necessity ofchanging mass to vary frequency and the difficultyof changing frequency and pre-load independently. The use of several specimens of varying geometry is a useful way of ensuring that data obtained are reliable and cover the appropriate range of parameters. Specimens should be simple and not have too low a height to diameter ratio to avoid errors resulting from highly non-uniform strains in the specimen. At very high frequencies, care should be taken to avoid standing waves in the specimens, which will render the above equations invalid. 3.2.
VIBRATING BEAM TESTS
The types of specimen used in the vibrating beam tests are illustrated in Figure 3. A typical block diagram is shown in Figure 4. In the tests, the beams were generally excited by means of a magnetic transducer at the free end and the response picked up by another magnetic transducer near the root. By using the graphic level recorder, response spectra comprising graphs of transverse peak velocity versus frequency were plotted and the resonant frequencies, f,, and half-power bandwidth, Aft, noted for many temperatures. For the high-temperature tests, however, for which suitable transducers were not available, the beam was instead excited by a shaker through an impedance head and a pillar made of Hastelloy-X [13]. In this case, the force transmitted back to the transducer from the beam was measured, with the force due to the pillar inertia cancelled electronically, and the same quantities, Jr, and Aft, measured. For reliable results with commercial versions of this method, it is usually necessary to use beams made of a relatively low modulus material, such as aluminum, with machined roots as indicated in Figure 3 for reduced root damping, not more than about 0.05 inches thick nor shorter than about 7 inches. However, for the enamels it was necessary to use another material because of the high temperatures involved and this made testing more difficult. The advantage of the method is that it is reasonably simple to use, errors can be assessed and kept within limits [12] and a wide band of frequencies and temperatures can be covered with a single specimen. The disadvantage is that only low-strain-level data are obtainable, within
DYNAMIC PROPERTIES OF DAMPING MATERIALS
455
#, Damp~ material
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Figure 4. Vibrating beam block diagram.
456
D.I.G. JONES
the linear range for the damping material. This, however, is not usually a problem since one would use another method if such data were needed. 3.2.1. "Oberst beam" tests For the beam with the damping material coated on one side only, the complex Young's modulus is derived from formulae [5, 11 ] due originally to Oberst. These are: 1 + 2en(2 + 3n + 2n 2) + e 2 n 4 z ~ =
,
1 +he
rl.._L= en [3+6n+4n2+2enS+ean" ~, r/D (1 + en) \ i ~ne'~Z'-+"3~n~3-+---~n 4 )
(7)
(8)
where Z 2 = (I + pDn/p)(f,[fo,) z, e = E r I E and n = h d h . in these formulae, Z 2 is calculated from the measured resonance frequency, f., of the nth mode of the damped beam and the measured frequency, fo,, of the undamped beam and e is then deduced from equation (7). t/o is then calculated from equation (8), this value of e and the measured value of t/, = Af,[f, being used. These equations give reasonably accurate results provided that Z 2 - 1 >10.1 or so [5, 11].
3.2.2. Modified "Oberst beam" tests For the beam with the damping material coated symmetrically on b0th-sides, the complex Young's modulus is derived [11] from the formulae Eo = E ( Z 2 - 1)/[8n 3 + 12n 2 + 6n], ,7o = ~ , Z ~ l ( Z
~ -
1),
(9) (10)
where Z 2 = (I + 2pon]p)(fdfo,) 2 and e and n are defined as before. In these formulae Z 2 and qs are determined as before. Again, the equation gives reasonably accurate results whenever Z 2 - I >i 0-1 or so [5, 11]. These formulae are also applicable directlyto the hightemperature specimens provided that the beam is coated symmetrically. A similar technique is also applicable for beams with multiple constrained layers added [14].
3.2.3. Sandwich beam tests Finally, for the symmetric sandwich beam, the approach is based on a set of equations due to Kerwin and others [15], reduced by the author and Nashif [12] to an uncoupled form. The shear modulus, GD, and loss factor, r/o, of the viscoelastic material may then be derived from the equations Go =
[(A - B) -- 2(a -- B) ~ - 2(Aq,) 21[~hh~, ~,~IL21 (1 -- 2A + 2B) 2 + 4(dq,) 2 '
(11)
rl'o = a q d [ A - B - 2(A - B) 2 - 2(Aq,)2],
(12)
A = (rodCOo,)2(2+ PD hJph)(B]2),
(13)
B = 1/6(1 + hD/h) 2.
(14)
where
and
DYNAMIC PROPERTIES OF DAMPING MATERIALS
457
For most elastomeric materials in the rubbery and transition regions, Eo ~ 3Go and
4. TEST RESULTS 4.1. FILLEDSILICONEELASTOMER[16] This material was evaluated by the resonance technique, square cross-section cylindrical specimens being used, with b = 0.50 in (12.7 mm) and ho = 0.86 in (21.8 mm) according to Figure I. Some of the measured values of ED and r/o for several temperatures and frequencies are shown in Table 1. To was taken arbitrarily as 45~ (505~ (281~ Graphs of ~r versus temperature are shown in Figure 5. Graphs of (To/T)ED and r/D versusf~r are plotted in Figure 6. The values of err were selected to give the best empirical fit of this data. It is seen that even the data at -100~ (200~ did not go far into the transition region, which is beyond l0 s Hz at 45~ (281~ as Figure 6 shows. The specimen link mass was 4.7 grams and the density was 0.048 lb/in 3 (1329 kg/m a) [17]. The data given here are limited to the linear region (max. strain ~< 10-3 in/in) for this particular material. Tests at higher strain levels were investigated also and showed non-linear amplitude dependent behavior. The results show that the data for many temperatures and frequencies can be collapsed on to a single curve and that the data obtained are sufficiently accurate and consistent for engineering purposes. 4.2. ACRYLICADHESIVE[18] This material was also evaluated by the resonance technique, square cross-section cylindrical specimens as illustrated in Figure 1 being used, with b = 0.525 and hD = 0.45 in. The mass of the link was 2.0 gm and the density of the material was 0.038 lb/in 3 (1052 kg/m3). Some measured test results for the tension--compression specimen are given in Table 2, for low input levels. The material is linear up to peak strain levels of well over I ~o, and so linearity was not an important factor in these particular tests. To was finally selected as 120~ (580~ 322~ so as to give a best fit to the experimental data and fit the previous value of~r. Graphs of(To/T)Eo and r/o versusf~r are plotted in Figure 7. Values ofct r are plotted versus temperature in Figure 5. It is seen from the results that Eo has a wide range of values and yet the scatter is reasonably small, even with r/o very large. The graphs of ~r versus temperature lie on the same curve as those for the filled silicone with this choice of To. 4.3. SILICONE POTTING COMPOUND 9
Again, the material properties were determined by a resonance technique, rectangular cross-section cylindrical specimens being used with bma~= 0"50 in (12-7 ram), bmi, = 0"45 in (11.4 mm) and hD = 1.00 in (25.4 mm). Measured results are given in Table 3. To was taken as 160~ (620~ 344~ Graphs of(To/T)Eo and 17D versusf~r are plotted in Figure 8. Values of a r are again plotted versus temperature in Figure 5, and again the curves of ~r versus temperature form a single curve. The link mass was 3.95 grams and the density was 0.038 lb/in 3 (1052 kg/m3). 4 . 4 . FILLED NITRILE RUBBER
[20]
This material was evaluated by the vibrating beam technique with the material coated symmetrically on both outer surfaces of the beam. The beam material was aluminum with a thickness of 0.032 inches and length L = 7 in. The damping material thickness was 0.I 19 in.
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Some of the measured values of Eo and ~Io for several temperatures and frequencies are shown in Table 4. To was taken as 195~ (655~ again on an empirical basis after trying several other values. The value To = 195~ gives a curve of~r versus temperature falls on the curves for other materials as Figure 5 shows. The composition of this particular material is given below, and further information on compounding and specimen preparation is given in references [20] and [21]. The density of the damping material was 0.041 lb]in ~ and that of the aluminum beam was 0.100 lb/in 3. The formulation of the particular sample investigated is as follows: Paracril-BJ 100 PHR SAF Black 50 Zinc Oxide I0 Sulfur 1.5 Benzothiozyl Disulfide 1.5 Anti-oxidant 1.0. ~05
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The specimen was milled and cured for 0.5 hours at a temperature of 320~ Graphs of. (To/T)ED and r/o versusf, r are shown in Figure 9. It will be noted from Table 4 that the specimen dimensions were such that Z ~/> 1-10 or so under all temperature and frequency conditions considered.
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4.5. POLYMERBLEND [21] The particular polymer blend tested here was a homogeneous milled mixture of ParacrilBJ, Paracril-D, Polyvinyl Acetate and Polystyrene, with the following specific formulation [20]: ParacriI-BJ Paracril-D Polyvinyl Acetate Polystyrene (Dylene 8) Zinc Oxide Anti-oxidant 2246 Dicumyl Peroxide SAF Black
50 P H R 100 50 100 10 1 7 50.
The specimens were cured at 280~ for 1 hour. Further details of specimen preparation are given in references [20] and [21]. The test specimens consisted of an aluminum beam 7 inches long, 0.5 inches wide and 0.032 inches thick, with a layer of the damping material 0.092 inches thick on each side for specimen A and 0.045 inches thick for specimen B. The density of the damping material was 0.041 lb]in 3 and that of the aluminum beam 0.100 lb]in s. The value of To was determined to be 230~ (690~ again on an empirical basis. Some of the reduced data for the tests on the two specimens is given in Table 5. Graphs of (To/T)Eo and ~/o versus foot are presented in Figure I0. Values of ~r as a function of temperature are plotted in Figure 5.
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The particular enamel tested [22] was applied as a slurzT to the two surfaces ofthe beam and fired at about I800~ The beam material was Hastelloy-X steel; which had a density of 0,286 Iblin 3 (7916 kglm3). Further coats were added as necessary and the beam re=fired until the thicknesses were uniform. Two beams were examined in these tests, although others were also tested [12], the first (A) being I,?5 inches (44.5 ram) long by 1.5 inches (38"I ram) wide by 0-030 inches (0.?6 ram) thick and having coating thickness ho = 0.0.185 inches (0.47 ram) and the other (B) 3.00 inches (76.2 ram) long by 0.5 inches 02"? ram) wide by frO? inches (I.78 ram) thick, and having a coating thickness ho = 0.0045 inches (0.11 ram). The density of the enamel was 0.09 Iblin 3 (2491 kglm3). Some of the observed data are recorded and reduced in Table 6. :Towas determined by trial and error to be 1250~ (1710~ 950~ Reduced data are plotted as graphs of (ToIT)ED and ~D versus'f0cr in Figure I I. Values of ccr are plotted v e r s u s temperature in Figure 5. )o7 : I =)l'"q
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5. UTILIZATION OF DATA The graph ofctr versus ( T - To), depicted in Figure 5, is very important because it represents a very general relationship which is accurate for a wide variety of materials. It can be shown, on the basis of a simple curve-fit, that ~r is given by Ctr -- [1 + ( T -- To)/2781-".
(15)
If, therefore, one is investigating the complex modulus data of any damping material and has test data in the form of blocks, each consisting of four numbers representing values of T, 9 f, Eo, and r/o, then one may semi-automate the process of reducing the data no matter how arbitrary the values and intervals of T a n d f m a y be. The general procedure which would have to be adopted is as follows: (1) make an initial guess for To; (2) compute Ctr from equation (15) with each T measured in degrees F at this point; (3) computefc~r and (To/T)Eo with To and T measured on the absolute scale at this point; (4) plot, by using a digital plotting routine, the graphs of (To/T)Eo and r/D versusfc~r; (5) from inspection of these graphs, select a new value of To and return to step (1); repeat until the best fitting data are obtained in step (4).
6. CONCLUSIONS
It has been shown in this paper that simple, well-established techniques are available for accurately measuring the complex modulus properties of damping materials. Provided that their limitations are recognized and respected, sufficient data can usually be obtained by use of each technique to completely characterize material damping behavior as a function of frequency and temperature over the range encountered in engineering applications. This is accomplished by conducting the tests over a wide range of temperatures and using the temperature-frequency equivalence principle to collapse the data to a single set of curves. The very general applicability of this principle has been verified for several dissimilar materials, ranging from a filled silicone elastomer through a polymer blend to a vitreous enamel, over a 550~ (300~ temperature range and a sixteen decade frequency range. This indicates that measurements obtained by using any reasonably accurate technique, even over a limited frequency range which would otherwise be of little value, can be collapsed for many temperatures to markedly extend the useful range of the measurement technique and thus obtain, on an engineering basis, data which have previously been obtainable only under more controlled laboratory conditions. Finally, the development of an approximate analytical relationship describing the variation of the shift factor with the temperature difference provides the basis for a semi-automatic process of reducing complex modulus data for damping materials no matter how arbitrarily the temperature and frequency values may be distributed, provided that the range is adequate. ACKNOWLEDGMENTS This work was sponsored by the Air Force Materials Laboratory, United States Air Force, under Project No. 7351, Task No. 735106. Thanks are due to A. D. Nashif, C. M. Cannon, R. W. Gordon and M. L. Parin for assistance with the tests, D. Gochoel and B. Dues for assistance with manuscript preparation and S. Askins for assistance with electronic instrumentation.
DYNAMICPROPERTIESOF DAMPINGMATERIALS
469
REFERENCES 1. J. D. FERRY 1961 Viscoelastic Properties of Polymers. New York: John Wiley and Sons, Inc. 2. J. D. FERRY, E. R. FITZGERALD,L. D. GRANDINE,JR. and M. L. WILLIAMS1952 Industrial and Engineering Chemistry44, 703-706. Temperature dependence of dynamic properties ofelastomers; relaxation distributions. 3. J. C. SNOWDON1968 Vibration and Shock hi Damped Mechanical Systems. New York: John Wiley and Sons, Inc. 4. B. J. LAZAN 1968 Damping of Materials and Members in Structural Mechanics. Oxford: Pergamon Press Ltd. 5. H. OBERST1952 Acustica (Akustische Beihefte) 4, 181-194. Ober die Dampfung Biegeschwingungen Dunner Bleche Durch fest Haftende Belage. 6. T. J. DtlDEK 1969 Journal of the Acoustical Society of America 46, 1384--1386. Damping material effectiveness measured by the Geiger-plate and composite-beam tests. 7. R.L. ADKINSJuly 1966ExperimentalAlechanics. Design considerations and analysis of a complex modulus apparatus. 8. J. L. EDWARDSand D. R. HICKS1972Journal ofthe Acoustical Society of America 52, 1053-1056. Useful range of a mechanical impedance technique for measurement of dynamic properties of materials. 9. D. I. G. JONESand M. L. PARtN 1972 Journal of Soundand Vibration 24, 201-210. Technique for measuring damping properties of thin viscoelastic layers. 10. C. M. CANNON,A. D. NASHIE,and D. I. G. JONES1968 Shock and Vibration Bulleth~ 38, 151-163. Damping measfirements on soft viscoelastic materials using a tuned damper technique. 11. A. D. NASHIF 1967 Shock and Vibration Bulletin 36, 37--47. A new method for determining damping properties of viscoelastic materials. 12. A. D. NASHIF1973 Shock and Vibration Bullethz 43, 145-151. Materials for vibration control in engineering. 13. A. D. NASHIF 1973 Proceedings of SEECO 73, Society of Electrical Engineers Conference, London. Damping of glass-like materials at high temperatures. 14. D. I. G. JONES,A. D. NASHIFand M. L. PARIN1973 Journal of Sound and Vibration 29, 423--434. Parametric study of multiple-layer damping treatments on beams. 15. D. Ross, E. E. UNGARand E. M. I~RWIN, JR. 1959 Structural Damping, American Society of Mechanical Engineers 49-87. Damping of plate vibrations by means of viscoelastic laminae. 16. BTR: a broad temperature range filled silicone elastomer supplied by Lord Mfg. Co., Erie, Pa. 17. 3M--467: an acrylic adhesive supplied by the Minnesota Mining and Manufacturing Company, Minneapolis, Minnesota. 18. SYI-~ARD188: a two part silicone RTV pottingcompound supplied by Dow Corning Corporation, Midland, Michigan. 19. PARACRm-BJ:a nitrile rubber formulation supplied by U.S. Rubber Company. 20. F. S. OWENS 1967 Shock and Vibration Bullethz 36, 25-35. Elastomers for damping over wide temperatures ranges. 21. F. S. OWENS1970 Air Force Materials Laboratory Report AFML-TR-70-242. Wide-temperature range free-layer damping materials. 22. VITREOUSENAMELCV 17214, manufactured by Chicago Vitreous Co., Chicago, Illinois. APPENDIX A b B D e E ED f fo. f. Go
NOMENCLATURE amplification factor I Y/-YI for resonance specimen; also a non-dimensional parameter breadth of cylindrical specimen non-dimensional parameter suffixdenoting damping material E~/E: the modulus ratio Young's modulus of beam material real part of complex Young's modulus of damping material frequency (Hertz) nth natural frequency of undamped beam nth resonant frequency of damped beam real part of complex shear modulus of damping material
470 h hD 1 L m M n
R S St T To X Z2 Ctr fl e r/D t/s tip ).n ~ p po
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D . I . C . JONES thickness of beam thickness of damping material length of viscoelastic material in shear resonance specimen length of beam mass of viscoelastic material added mass on resonance test specimen hD/h: thickness ratio; also suffix indicating mode number radius of gyration of shear specimen about horizontal axis in Figure I (R = l/~/i2for rectangular specimen) load carrying area of damping material in resonance specimens non-load carrying (free) area of damping material in resonance specimens temperature reference temperature input acceleration for resonance specimen output acceleration for resonance specimen non-dimensional parameter temperature shift factor shape factor for cylindrical resonance test specimen; fl = 2.0 for unfilled clastomers and ~ 1-5 for filled elastomers strain shear loss factor loss factor of beam specimen extensional loss factor of damping material wavelength ofnth beam mode nth eigenvalue for beam density of beam; also density in general density of damping material where specific identification is necessary density at reference temperature stress circular frequency nth circular frequency of damped beam nth circular frequency of undamped beam