Dynamic stiffness of ageing rubber vibration isolators Leif Kari
Structure-borne sound Source Receiver
Physical principle Hard Hard Soft
F without F with Force transmissibility T F = F with F without
F e Ideal isolator k m u e 0 ω 2 mu e = F e - F with F with = k u e F with k F with T F = Fwith F without 1 = ω 1 2 ω 2 0 ω 02 = k /m
Rigid foundation ideal isolator 10 2 10 0 Force transmissibility 10 2 10 4 10 6 10 8 No isolator Ideal isolator 10 0 10 1 10 2 10 3 10 4 Frequency [Hz]
Rigid foundation ideal isolator 10 2 10 0 Force transmissibility 10 2 10 4 10 6 10 8 No isolator Ideal isolator 10 0 10 1 10 2 10 3 10 4 Frequency [Hz] η
Nonrigid foundation F without F with
Foundation stiffness F f F f u f h f u f k f = F f / u f k f = h 2 E iω8 f 12(1 2 ) f ρ f ν f
Nonrigid foundation Ideal isolator 10 2 10 0 Force transmissibility 10 2 10 4 10 6 10 8 No isolator Nonrigid foundation Rigid foundation 10 0 10 1 10 2 10 3 10 4 Frequency [Hz]
Acoustic radiation Wave fronts Wall 1 W/m 2 120 db!
ideal isolator non-ideal isolator m m F ideal F in k u in uout F ideal F out
ideal isolator non-ideal isolator F ideal F in k u e u in uout F ideal F out F ideal = k u e F in = k inin u in + k inout u out F out = k outin u in + k outout u out with k inout = k outin
Constitutive preliminaries Spherical part tr =3 (,, )div Deviatoric part dev =2 (,, ) dev + (,, ; ) dev ( ) d lim t µ = µ lim t µ 1 = 0 [Kari 2016a,b]
Equilibrium elastic modulus,, =,, Density α= 1 ρ ρ ρ T (equlibrium) (1 α T)ρ 0 T = T T 0
Specific relaxation function,, ; = h( ) Non-dimensional relaxation intensity» 1 = Γ(1+β ) 0 < β 1 =10 [Kari 2016a,b]
Physical ageing [Cangialosi et al Soft Matter 2013]
Physical ageing cont
Modelling physical ageing = = lim t = d d = = exp = + = [Greiner & Schwarzl 1984, Kovacs 1963, Doolittle 1953, Cohen & Turnbull 1959]
Modelling physical ageing modified d d = D = D = 1 Γ(1 ) 1 ( ) d( ( )) d d = exp = 10 = log e = 0.434294 [Kari 2016a,b]
WLF shift function =10 = = 10 = 10 = + = = [Greiner & Schwarzl 1984, Kovacs 1963, Doolittle 1953, Cohen & Turnbull 1959]
Cont = = =10 = =,, ; = h( ) [Kari 2016a,b]
Modelling chemical ageing Scission of polymer chains, = 1 D =1 = e =1
Modelling chemical ageing cont Plus reformation of new polymer links, = 1 + D =1 = e =1 [Kari 2016a,b]
Modelling chemical ageing cont Scission and reformation of new polymer links, = + 1 [Kari 2016a,b]
Vibration isolator [Kari et al. 2001]
ideal isolator non-ideal isolator F ideal F in k u e u in uout F ideal F out F ideal = k u e F in = k inin u in + k inout u out F out = k outin u in + k outout u out with k inout = k outin
Modelling approaches - Wave-guides Infinite beam Traction free surface Wave equations Bessel Exp. harm. Trig. Satisfy traction free B.C:s Dispersion relation [Kari 2001a,b, Östberg et al. 2011]
Nonrigid foundation Real isolator 10 2 10 0 Force transmissibility 10 2 10 4 10 6 10 8 No isolator Real isolator Ideal isolator Ideal isolator Rigid foundation 10 0 10 1 10 2 10 3 10 4 Frequency [Hz]
DMTA measurements and modelling [Kari et al. 2001]
Cont Transfer Stiffness [N/m] a) 10 7-60ºC 10 6 +60ºC 10 5 0ºC -25ºC 10 4 +25ºC 10 3 10 1 10 2 10 3 10 4 b) Driving Point Stiffness [N/m] 10 6 10 4 10 2-60ºC -25ºC 10 1 10 2 10 3 10 4 Frequency [Hz] 0ºC +25ºC +60ºC [Kari et al. 2001]
References Cangialosi, D., Boucher, V.M., Alegria, A., Colmenero, J.: Physical aging in polymers and polymer nanocomposites: recent results and open questions. Soft Matter 9, 8619 8630 (2013) Cohen, M.H., Turnbull, D.: Molecular transport in liquids and glasses. J. Chem. Phys. 31, 1164 1169 (1959) Doolittle, A.K.: Studies in newtonian flow. II. The dependence of the viscosity of liquids on free-space. J. Appl. Phys. 22, 1471 1475 (1951) Greiner, R., Schwarzl, F.R.: Thermal contraction and volume relaxation of amorphous polymers. Rheol. Acta23, 378 395 (1984) Kari, L.: On the waveguide modelling of dynamic stiffness of cylindrical vibraitnoso iltaors. Part I: The model, solution and experimental comparison. J. Sound. Vib. 244, 211 233 (2001a) Kari, L.: On the waveguide modelling of dynamic stiffness of cylindrical vibration isolators. Part I: The dispersion relation solution, convergence analysis and comparison with simple models. J. Sound. Vib. 244, 235 257 (2001b) Kari, L.: Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range. Part 1: Constitutive equations. Continuum Mech. Thermodyn. Submitted (2016a) Kari, L.: Dynamic stiffness of chemically and physically ageing rubber vibration isolators in the audible frequency range. Part 2: Waveguide solution. Continuum Mech. Thermodyn. Submitted (2016b) Kari, L., Eriksson, P., Stenberg, B.: Dynamic stiffness of natural rubber cylinders in the audible frequency range using wave guides. Kaut. Gummi Kunstst. 54, 106 111 (2001) Kovacs, A.J., Aklonis, J.J., Hutchinson, J.M., Ramos, A.R.: Isobaric volume and enthalpy recovery of glasses. II. A transparent multiparameter theory. J. Polym. Sci., Part B: Polym Phys 17, 1097 1162 (1979) Odegard, G.M., Bandyopadhyay, A.: Physical aging of epoxy polymers and their composites. J. Polym. Sci., Part B: Polym Phys49, 1695 1716 (2011) Östberg, M., Kari, L.: Transverse, tilting and cross-coupling stiffness of cylindrical rubber isolators in the audible frequency range the wave-guide solution. J. Sound. Vib. 330, 3222 3244 (2011)