BS 7544-1992 Method for graphical presentation of the complex modulus of damping materials《阻尼材料复数模数的 图形表示方法》.pdf

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1、BRITISH STANDARD BS 7544:1991 ISO 10112: 1991 Method for Graphical presentation of the complex modulus of damping materialsBS7544:1991 This British Standard, having been prepared under the directionof the General Mechanical Engineering Standards Policy Committee, waspublished under the authorityof t

2、he Standards Boardand comes into effect on 31 January1992 BSI 10-1999 The following BSI references relate to the work on this standard: Committee reference GME/21 Draft for comment 89/79060 DC ISBN 0 580 20401 4 Committees responsible for this British Standard The preparation of this British Standar

3、d was entrusted by the General Mechanical Engineering Standards Policy Committee (GME/-) to Technical Committee GME/21, upon which the following bodies were represented: Cranfield Institute of Technology Department of Trade and Industry (National Engineering Laboratory) Electricity Supply Industry i

4、n England and Wales Engineering Equipment and Materials Users Association Health and Safety Executive Institute of Sound and Vibration Research Institution of Mechanical Engineers Lloyds Register of Shipping Ministry of Defence Power Generation Contractors Association (BEAMA Ltd.) Society of British

5、 Aerospace Companies Ltd. Society of Environmental Engineers Society of Motor Manufacturers and Traders Ltd. University of Manchester Amendments issued since publication Amd. No. Date CommentsBS7544:1991 BSI 10-1999 i Contents Page Committees responsible Inside front cover National foreword ii Intro

6、duction 1 1 Scope 1 2 Nomenclature 1 3 Graphical presentation 6 Annex A (informative) Bibliography Inside back cover Figure 1 Data quality check 4 Figure 2 Temperature shift function and its properties 5 Figure 3 Complex modulus reduced frequency plot 6 Figure 4 Complex modulus inverted-“U” plot 7 T

7、able 1 Complex modulus data 2 Table 2 Complex modulus data as a function of temperature and frequency 3 Table 3 Example of analytical representation of the temperature shift function 8 Table 4 Example of analytical representation of the complex modulus 8BS7544:1991 ii BSI 10-1999 National foreword T

8、his British Standard has been prepared under the direction of the General Mechanical Engineering Standards Policy Committee. It is identical with ISO10112:1991 “Damping materials Graphical presentation of the complex modulus”, which was prepared by Technical Committee ISO/TC108 of the International

9、Organization for Standardization (ISO) and in the development of which the United Kingdom played an active part. A British Standard does not purport to include all the necessary provisions of a contract. Users of British Standards are responsible for their correct application. Compliance with a Brit

10、ish Standard does not of itself confer immunity from legal obligations. Summary of pages This document comprises a front cover, an inside front cover, pages i and ii, pages1 to 8, an inside back cover and a back cover. This standard has been updated (see copyright date) and may have had amendments i

11、ncorporated. This will be indicated in the amendment table on the inside front cover.BS7544:1991 BSI 10-1999 1 Introduction Damping is one potential approach to reducing vibration levels in a structural system. Damping is the dissipation of vibratory energy by converting it into heat, as distinguish

12、ed from transporting it to another part of the system. When the damping is due to internal energy dissipation within a material which is part of the structural system, and when the damping is of engineering significance, the material is called a vibration damping material. The energy dissipation is

13、due to molecular or crystal-lattice interactions and can be measured in terms of the stress/strain hysteresis loop of the vibration damping material. Other possible sources of damping, such as plastic deformations in the joints, relative slip at joints, air pumping in the joints, acoustic radiation

14、of energy, eddy current losses, etc., are not covered in this International Standard. The mechanical properties of most damping materials depend on frequency, temperature and strain amplitude at large strains; since this International Standard is restricted to linear behaviour, it does not cover the

15、 strain amplitude effect. 1 Scope This International Standard establishes the graphical presentation of the complex modulus of viscoelastic vibration damping materials which are macroscopically homogeneous, linear and thermorheologically simple. The complex modulus may be the shear modulus, Youngs m

16、odulus, bulk modulus, longitudinal wave propagation modulus, or Lam modulus. This graphical presentation is convenient and sufficiently accurate for many vibration damping materials. The preferred nomenclature (parameters, symbols and definitions) is also given. The primary purpose of this Internati

17、onal Standard is to improve communication among the diverse technological fields concerned with vibration damping materials. 2 Nomenclature 2.1 Complex modulus The operator form of the constitutive equation for the linear, isothermal, isotropic, macroscopically homogeneous, thermorheologically simpl

18、e seeequation (7) viscoelastic material being deformed in shear is defined 1 as where The operator p Ris defined as The reduced time differential dt Ris defined as where The Fourier transform (f.t.) of equation (1) leads to the definition of G, the complex shear modulus valid for steady-state sinuso

19、idal stress and strain, as where * (j R ) denotes the f.t. of (t). The reduced circular frequency, R , is given as which is a product of , the circular frequency, in radians per second, and the dimensionless temperature shift function, while f Rand f denote the reduced cyclic frequency and the cycli

20、c frequency, in hertz. The complex shear modulus is dependent on both frequency and temperature If (and only if) the dependency is expressed as then the material is called thermorheologically simple (TRS). Furthermore, equations (1) to (7) apply only to linear conditions. Alternatively, consider a v

21、iscoelastic material element which undergoes a sinusoidal shear strain3 which lags the sinusoidal shear stress by the phase angle G : P(p R ) (t) = Q(p R ) (t) . . . (1) (t) is the shear stress; (t) is the shear strain; P(p R ) and Q(p R ) are polynomials in p R . p R= d/dt R . . . (2) dt R= dt/ T (

22、T) . . . (3) t is time, in seconds; T (T) is the dimensionless temperature shift function 2 dependent on temperature, T, in kelvin. . . . (4) . . . (5) G = G(, T) . . . (6) G = G(j R ) = Gj T (T) . . . (7) = Asin t . . . (8) = Asin(t + G ) . . . (9)BS7544:1991 2 BSI 10-1999 The sinusoidal strain and

23、 stress may be represented in complex notation as The complex shear modulus, G, may be equivalently defined as where The concept holds for one-, two- and three-dimensional states of stress and strain 2. Developments similar to the above apply to Youngs modulus, E, to the bulk modulus, K, to the Lam

24、modulus, , and to the longitudinal wave propagation modulus W =+2G. A thermorheologically simple material is a material for which the complex modulus may be expressed as a complex valued function of one independent variable, namely reduced frequency, to represent its variation with both frequency an

25、d temperature. NOTE 1Sometimes, the real modulus and the material loss factor are treated as independent functions of reduced frequency; while this can facilitate satisfactory applications, it is a conceptual error. The complex modulus evaluated at a given temperature and a given frequency represent

26、s both the magnitude and the phase relationships between sinusoidal stress and strain. 2.2 Data check It is presumed in this International Standard that a set of valid complex modulus data (e.g., Table 1 andTable 2) has been obtained in accordance with good practice (see, for example, ref. 4). It is

27、 recommended that each set of data be routinely and carefully scrutinized. As a minimum, the lg Gversus lg G Mshould be plotted (e.g., Figure 1). If the set of data represents a thermorheologically simple material, if an adjustment of modulus for temperature and density is not appropriate and if the

28、 set of data has no scatter, the set of data will plot as a curve of vanishing width. Each point along the arc of the curve corresponds to a unique value of reduced frequency see equation (6). However, this is not considered in this plot. The material loss factor and the modulus magnitude are cross-

29、plotted, and the reduced frequency, temperature, and frequency parameters do not occur explicitly. No part of any scatter in this plot can be attributed to an imperfect temperature shift function. The loss factor versus modulus magnitude logarithmic plot can reveal valuable information regarding sca

30、tter of the experimental data. The width of the band of data, as well as the departure of individual points from the centre of the band, are indicative of scatter. Acceptable scatter depends on the application. Nothing is revealed about the accuracy of the temperature and frequency measurements or a

31、bout any systematic error. Table 1 Complex modulus data . . . (10) . . . (11) . . . (12) G M is the magnitude of the shear modulus; G R= G is the real (storage) modulus; G l= G = G R G is the imaginary (loss) modulus; G= tan G is the material loss factor in shear. Tmodel NALF 0 NA 0 A(1) A(2) A(3) A

32、(4) A(5) A(6) Complex modulus model NVEM 11 NB 9 B(1) 5,70 B(7) 0,410 B(2) 212 B(8) 0,257 B(3) 176 B(9) 3,65 B(4) 0,662 B(10) B(5) 4,510 10 2 B(11) B(6) 3,000 10 2 B(12)BS7544:1991 BSI 10-1999 3 Table 2 Complex modulus data as a function of temperature and frequency Temperature Frequency G R G G l T

33、 (T) (K) (Hz) (MPa) (MPa) 254,2 7,800 244,0 0,130 0 31,72 2,795 6 10 4T 254,2 15,60 252,0 0,114 0 28,73 2,795 6 10 4T 254,2 31,20 260,0 9,970 0 10 2 25,92 2,795 6 10 4T 254,2 62,50 266,0 9,410 0 10 2 25,03 2,795 6 10 4T 254,2 125,0 275,0 9,470 0 10 2 26,04 2,795 6 10 4T 254,2 250,0 281,0 7,210 0 10

34、2 20,26 2,795 6 10 4T 254,2 500,0 292,0 8,160 0 10 2 23,83 2,795 6 10 4T 254,2 1 000 337,0 7,030 0 10 2 23,69 2,795 6 10 4T 273,2 7,800 77,30 0,623 0 48,16 33,94 T 273,2 15,60 96,50 0,541 0 52,21 33,94 T 273,2 31,20 119,0 0,456 0 54,26 33,94 T 273,2 62,50 140,0 0,385 0 53,90 33,94 T 273,2 125,0 162,

35、0 0,333 0 53,95 33,94 T 273,2 250,0 185,0 0,259 0 47,92 33,94 T 273,2 500,0 206,0 0,232 0 47,79 33,94 T 273,2 1 000 242,0 0,202 0 48,88 33,94 T 283,2 7,800 22,90 0,892 0 20,43 2,825 T 283,2 15,60 30,70 0,869 0 26,68 2,825 T 283,2 31,20 42,10 0,813 0 34,23 2,825 T 283,2 62,50 57,90 0,731 0 42,32 2,82

36、5 T 283,2 125,0 77,30 0,645 0 49,86 2,825 T 283,2 250,0 101,0 0,532 0 53,73 2,825 T 283,2 500,0 126,0 0,461 0 58,09 2,825 T 283,2 1 000 161,0 0,388 0 62,47 2,825 T 292,2 7,800 11,10 0,745 0 8,270 0,644 6 T 292,2 15,60 14,00 0,818 0 11,45 0,644 6 T 292,2 31,20 18,30 0,869 0 15,90 0,644 6 T 292,2 62,5

37、0 24,80 0,889 0 22,05 0,644 6 T 292,2 125,0 34,60 0,875 0 30,27 0,644 6 T 292,2 250,0 48,90 0,786 0 38,44 0,644 6 T 292,2 500,0 68,00 0,703 0 47,80 0,644 6 T 292,2 1 000 94,80 0,611 0 57,92 0,644 6 T 303,2 7,800 7,800 0,552 0 4,306 0,194 1 T 303,2 15,60 9,560 0,661 0 6,319 0,194 1 T 303,2 31,20 12,1

38、0 0,762 0 9,220 0,194 1 T 303,2 62,50 16,10 0,856 0 13,78 0,194 1 T 303,2 125,0 22,30 0,912 0 20,34 0,194 1 T 303,2 250,0 30,70 0,874 0 26,83 0,194 1 T 303,2 500,0 45,60 0,824 0 37,57 0,194 1 T 303,2 1 000 65,40 0,747 0 48,85 0,194 1 T 313,2 7,800 6,000 0,351 0 2,106 4,308 8 10 2T 313,2 15,60 6,800

39、0,448 0 3,046 4,308 8 10 2T 313,2 31,20 7,940 0,553 0 4,391 4,308 8 10 2T 313,2 62,50 9,520 0,661 0 6,293 4,308 8 10 2T 313,2 125,0 11,80 0,775 0 9,145 4,308 8 10 2T 313,2 250,0 15,30 0,845 0 12,93 4,308 8 10 2T 313,2 500,0 20,80 0,897 0 18,66 4,308 8 10 2T 313,2 1 000 30,20 0,894 0 27,00 4,308 8 10

40、 2TBS7544:1991 4 BSI 10-1999 Table 2 Complex modulus data as a function of temperature and frequency Temperature Frequency G R G G l T (T) (K) (Hz) (MPa) (MPa) 333,2 7,800 5,000 0,110 0 0,550 0 5,083 1 10 3T 333,2 15,60 5,200 0,152 0 0,790 4 5,083 1 10 3T 333,2 31,20 5,480 0,214 0 1,173 5,083 1 10 3

41、T 333,2 62,50 5,890 0,289 0 1,702 5,083 1 10 3T 333,2 125,0 6,450 0,394 0 2,541 5,083 1 10 3T 333,2 250,0 7,280 0,506 0 3,684 5,083 1 10 3T 333,2 500,0 8,690 0,601 0 5,223 5,083 1 10 3T 333,2 1 000 11,00 0,657 0 7,227 5,083 1 10 3T 353,2 7,800 4,950 3,220 0 10 2 0,159 4 7,650 0 10 4T 353,2 15,60 5,0

42、10 4,510 0 10 2 0,226 0 7,650 0 10 4T 353,2 31,20 5,090 7,000 0 10 2 0,356 3 7,650 0 10 4T 353,2 62,50 5,210 0,101 0 0,526 2 7,650 0 10 4T 353,2 125,0 5,340 0,158 0 0,843 7 7,650 0 10 4T 353,2 250,0 5,620 0,236 0 1,326 7,650 0 10 4T 353,2 500,0 6,070 0,317 0 1,924 7,650 0 10 4T 353,2 1 000 7,240 0,2

43、88 0 2,085 7,650 0 10 4T Figure 1 Data quality checkBS7544:1991 BSI 10-1999 5 2.3 Temperature shift function The set of complex modulus data itself implicitly defines the temperature shift function T (T), provided the experimental ranges of temperature and frequency are adequate. It is assumed that

44、a single temperature shift function is applicable. It is recommended that the following three functions be plotted for the experimental range of temperature (e.g., Figure 2) because a) the temperature shift function, T (T), has historically had a central role; b) its slope, d(lg T )/dT, is the cruci

45、al feature that causes data to be correctly shifted; and c) the apparent activation energy 2, %H A , is of interest and is given by where R is the gas constant %H A= 2,303RT 2 d(lg T )/dT . . . (13) R = 0,00828 Nkm/gmolK . . . (14) Figure 2 Temperature shift function and its propertiesBS7544:1991 6

46、BSI 10-1999 3 Graphical presentation 3.1 Reduced frequency plot A set of complex modulus data is presented in Figure 3. A logarithmic scale is shown along the vertical axis for the real and the imaginary modulus components, in megapascals (MPa), and for the dimensionless loss factor. The logarithmic

47、 scale along the horizontal axis is reduced cyclic frequency, f R , in hertz. The reduced frequency for the i thexperimental point, f Ri , is given by where 3.1.1 Jones temperature lines In Figure 3 the vertical logarithmic scale on the right is the cyclic frequency, in hertz. The non-uniformly spac

48、ed diagonal constant temperature lines, together with the horizontal reduced frequency axis and the vertical frequency axis, provide a temperature-frequency-reduced frequency nomogram 5. The logarithmic form of equation (5) is which is the equation for the straight line inFigure 3. Values of tempera

49、ture, in kelvin, at convenient intervals are chosen. The spacing of the set of constant temperature lines depends on the temperature shift function used. The range of the diagonal lines should be chosen to be the same as the experimental temperature range of data to preclude unintentional (and possibly highly erroneous) extrapolation. Fu