Torsional damper for maximum energy absorption with equilibrated polydimethylsiloxanes as damping fluids

Journal of Sound and Vibration (1982) 82(4), 465-472

TORSIONAL

DAMPER

FOR MAXIMUM

WITH EQUILIBRATED

ENERGY

ABSORPTION

POLYDIMETHYLSILOXANES

AS DAMPING

FLUIDS

R. ANDRE AND J. H. SPURK Technische Hochschule Darmstadt, Technische Striimungslehre, 06100

Darmstadt, Germany

(Received 3 August 1981)

Shear viscosity and effective shear modulus, quantities related to the complex viscosity, have been measured as functions of frequency for five polydimethylsiloxanes commonly used as damper fluids. Maximum energy dissipation is obtained by realizing a damper whose damping constant times the shear viscosity divided by the product of effective shear modulus and moment of inertia of the inertia member equals one. Experiments show that in this tuning the dissipated energy when polydimethylsiloxanes are used as damping fluids can be as much as a factor of two higher than the maximum dissipated energy when using Newtonian fluid.

1. INTRODUCTION operating reciprocating engines over a wide speed range it is usually impossible to avoid critical speeds and an artificial damper is applied to provide the necessary energy dissipation. In the commonly used Houdaille damper, a flywheel is mounted in an oiltight case with small clearances and the case is filled with silicone oil (polydimethylsiloxane) [l]. The theory of these untuned dampers has been described in textbooks: e.g., in reference [2]. In the theory the assumption is made that silicone oil behaves as a Newtonian fluid. The important result of this theory is the condition for maximum energy dissipation: i.e., c = IDw, where c is the damping constant, which is equal to the shear viscosity of the fluid times a geometric factor [ 11, and where IO is the moment of inertia of the damper flywheel

When

and w is the radian frequency. Michele [3] has shown that silicone oil behaves like a second order fluid in steady shear flow, up to shear rates where shear stresses are comparable to the first normal stresses. In oscillatory shearing flow, of the kind occurring in the fluid-filled clearances of a damper, the non-Newtonian behaviour of the fluid must manifest itself and this will lead to a different condition for optimal damping. Spurk has given the theory of torsional dampers in fluids of second order [4]. The extra stress T of this fluid is given by T = vA1 + PA: + yA,, where Al and A2 are the first and second Rivlin-Eriksen tensors.

Only two of the three material functions n, /? and y enter the expression for dissipated energy, in the form of a complex viscosity n x = n + iwy (y < 0). These material functions are constants for a second order fluid and Spurk has shown that a frequency independent tuning for optimal damping is then possible, with values of dissipated energies considerably different from those computed on the basis of Newtonian fluid. From Michele’s steady shear experiments, one can conclude that silicone oil will behave as a fluid of second order also in oscillatory flow, with constant n and y up to radian frequencies w at which -WY becomes comparable with the shear viscosity. For higher frequencies, and these are of interest in damper applications, one expects a dependence of the material functions on frequencies. Spurk’s analysis applies also to “generalized” fluids of second order, where 465 0022-460X/82/120465+08

$03.00/O

@ 1982 Academic Press Inc. (London) Limited

466

R. ANDRli

AND J. H. SPURK

these functions depend on frequency and also to linear viscoelastic fluids or, more generally, to general viscoelastic fluids, at least for sufficiently small amplitudes. For frequency dependent y, the tuning is also frequency dependent and a very appealing feature of second order fluids is lost. In this paper measurements are reported of 77 and y as functions of frequency for polydimethylsiloxanes commonly used in damper applications, the energy dissipation which can be obtained with a given set of functions y(o) and g(w) of a particular fluid is established and the sensitivity of the tuning to the frequency dependence of y(o) is shown. It is also shown that the design of dampers can be based, with accuracy sufficient for engineering application, on the assumption of average but constant values of 77 and y.

2. DISSIPATED

ENERGY

Figure 1 shows a torsional vibration damper, consisting of the casing, which is to be attached to the vibrating structure and a loose flywheel of moment of inertia ID. The liquid in the clearances (Si, &) is sheared whenever a relative motion occurs between casing and the inertia member and energy is then dissipated in the fluid.

Figure 1. Torsional vibration damper.

If the inertia of the fluid and “corner” effects are neglected (which is possible in most applications), one finds from reference [4] the work dissipated per cycle WE

Here %’is the work W Newtonian fluid [2]; & member relative to the rl and r2 and the axial

2(X77/bW)

-2x77R2

(xrllbd2+(1 +xrm2 - bJ

(1 dJG .

referred to the maximum dissipated work W,,, = (7r/2)IDw2c5$ in a is the amplitude of the casing and & is the amplitude of the inertia casing. The geometric factor x is, if the two radial surfaces between surface are wetted (as in Figure l), given by x =

(&SJl

-

(rl/r2)4+2(LSIIr2S2)1.

(2)

All operating points of viscous dampers lie on the surface I!’ = f(&(&), yx/ID) shown in Figure 2, regardless of whether n or y are frequency dependent or not. Curve 1 satisfies the condition y,y/ID + 1 = 0 and gives, for fixed values of xn/(IDo), the largest dissipated energy, as is also evident from equation (1). For a second order fluid, this tuning is frequency independent, but it is worth pointing out that this is so even when n is frequency dependent. This curve incidentally, is also the envelope for the damping of all tuned

VIBRATION

Figure 2. The dissipated fluid, y = 0.

DAMPER

work @. 0, Curve for optimal

WITH

SILICONE

damping,

OIL

yx/ID + 1 = 0; 0, damping

467

for a Newtonian

dampers, for here one has

where k is the spring constant. This tuning is of course frequency dependent. Curve 2 shows the damping for a Newtonian fluid (y = 0); the same damping is obtained for the tuning yx/ID = -2, since w is double valued, as the projection of w in the xn/(Im) direction shows. For large values of xn/(IDw) there is no difference in damping between Newtonian and viscoelastic fluids, but substantial differences occur for smaller xn/(IDw): i.e., higher or lower values of dissipated energy can be achieved depending on the value of yx/ID. The operational points for a given damper in a second order fluid move along curves on the surface m whose projections in the xn/(IDu), -&ID-plane are straight lines at yx/ID = const. For a general viscoelastic fluid, for which q(w) and y(o) are functions of frequency, the operational points move along curves whose projections in the x~/Igw, yx/ID-plane follow by elimination of w from these two functions. If the curve is single valued the optimal damping yx/ID = - 1 is reached at only one point; the location of this point depends on the material properties of the particular fluid. For a Maxwell fluid, one has, for the complex viscosity, nx = n’-iv”, with n’ = no/[1 +(wr,J’] and q”= nOw~,,/[l +(oQ)*], where no and r. are the constants of the Maxwell fluid. Replacing, in equation (l), n by n’ and --WYby 7” gives the dissipated work w for a Maxwell fluid. Figure 3 shows w as a function of xq’/IDo with xqoro/ID as parameter for this fluid (projection of the Maxwell-curve lying on the surface I@ in the negative y,y/ID direction). Also shown is the curve for optimal damping and the curve for Newtonian fluid.

468

R. ANDRa

Figure 3. The dissipated work 6’. 0, fluid; 3a, , damping for a Maxwell fluid:

c3

AND

J. H. SPURK

Curve for optimal @, xvoro/ZD = 1;

3. THE

damping;

0,

damping

for

a Newtonian

@, ,YJOTO/ID = 5.

RHEOMETER

The damper shown in Figure 1 may immediately be used to determine experimentally the functions n(o) and y(w). From equation (1) one has, for the amplitude ratio, &I&

= {(xn/IDo)2 + (1 +xrlM2~~“2,

(3)

and measurement of this amplitude ratio with a given value x/ID = ai at a fixed frequency, and another measurement at the same frequency with another value x/ID = u2, give with expression (3), two equations for the two unknowns n(w) and y(w), i =

(~)R/~G);2=Oi=~?{77(~)l~}2+{1+air(~)}2,

1,2,

(3a)

with the solutions l/2

9 y(w)

=

_1(@2-

2

l)(aJaz)-

(@1a2-a1

l)b2/ad

(4) (5)

In addition to the frequency w and the geometric quantity x/ID only the ratio of two amplitudes is needed. These quantities can be measured quite accurately. The damper has no free surfaces which, in other types of rheometers, often lead to troublesome complications. Figure 4 shows measured values of n(w) and y(w), the latter in the form of the storage with molecular modulus G’(w) = - y(w)w2, for five equilibrated polydimethylsiloxanes weights from 10 000 to 100 000 at temperature T = 22°C. For low frequencies the shear viscosity n is almost constant, but decreases with higher frequencies. This decrease occurs for silicone oils with high molecular weights at lower frequencies than for oils with lower molecular weights. The values of G’(w) are nearly proportional to w2, up to frequencies where the decrease of shear viscosity becomes evident, indicating constant y. The value of G’(w) at these

VIBRATION

DAMPER

. I Al

100

IO'

.

WITH

SILICONE

469

OIL

??

I

I I lllll

I

I

111’lI’

I I IO2

IO'

IO' w

(s-‘I

polydimethylsiloxanes Figure 4. Experimental values of q(o) and G’(w) = - y( w ) 0’ for five equilibrated ; 20 000, V, v; 30 000, 0, 0: 50 000, a, (T = 22°C). Silicone molecular weight M, II, G’: 10 000, cl, ?? 100 000, 0, ?? .

A;

frequencies is about half the first normal stress difference measured by Michele [3]. At higher frequencies the increase of G’(w) is less than w*. Michele has defined, from the constants n and y of a second order fluid, the average relaxation time 7 = - r/n = n/i=, where F is interpreted as an effective shear modulus. In his measurements i; was found to be nearly independent of molecular weight (I; = 1.28 x lo4 N/m2), and his values for n agree with our data at low frequencies.

Figure 5. The dimensionless viscosity M 100 000. For ~(0) see Figure 4.

~(w)/~(O).

0, M 10 000; V, M 20 000; 0, M 30 000; &M

50 000; 0,

In Figure 5 the viscosity ratio n (w)/n (0), where n (0) is the viscosity at zero frequency, is shown versus the frequency made dimensionless by using this average relaxation time. All the measured data for five different silicone oils, of molecular weights differing by as much as a factor of ten, then fall on one curve. The ratio -~‘(W)/?(W) corresponds to the effective modulus r and will be subsequently so called. It is shown, made dimensionless with Michele’s value, in Figure 6. At low frequencies Michele’s value is reached; at high frequencies - n*(w)/r(w) is constant and is about twice as large as Michele’s value.

470

R. ANDRii

AND

J. H. SPURK

IL 3

x

,‘ -

I

3 C

0.7

Figure6. Thedimensionlesseffectiveshearmodulus A, M 50 000; 0, M 100 000. r= 1.28 x lo4 N/m*.

-q2(w)/y(w)f.0,A4

10 000; V,M20 OOO;O,M 30 000;

Both n and y are temperature dependent and it is to be expected that this dependence manifests itself mostly through a change in the zero frequency values of n and y and therefore through a change in the relaxation time. In the absence of data for the temperature effects in oscillatory shear flow, the temperature dependence may be inferred from Michele’s data in steady shear flow; he found that the effective modulus is nearly independent of temperature. It should also be mentioned that no amplitude dependence of 77and y was noticed in our experiments and no degradation with time. Figures 5 and 6 give the necessary data for the layout of a damper. For a given frequency the layout can be done by treating the viscoelastic fluid as a second order fluid, since in the neighbourhood of this frequency all viscoelastic fluids behave locally as second order fluids. This amounts to basing the layout on average values n,, and r,, in the frequency range of interest. 4. DAMPER

PERFORMANCE

In order to exhibit the effects of frequency dependence of the fluid properties on the performance, in this section a comparison is made between theory and experiments for a given damper, whose design is based on an average value of shear viscosity naV= O-8 n (0) and an effective shear modulus of r,, = l-9 x lo4 N/m*. Thus, the theoretical curves apply to second order fluids. In Figure 7 the dissipated energy, which has been computed from the measured amplitude ratio &/c#I~ according to equation (l), is compared with predictions. The experimental points have been obtained up to values of the reduced frequency WT= 2, where r is Michele’s relaxation time for the particular fluid [3]. Agreement between measured and computed energy is very good for Newtonian flow, which gives an indication of the accuracy of the measurements. The experimental points for the polydimethylsiloxanes follow the theoretical curves for a second order fluid quite closely. Deviations occur at high frequencies, where the frequency dependence of y(w) becomes pronounced. Even though the measurements extend to the same reduced frequency range ~7, the deviations occur earlier for intermediate values of the tuning parameter qtvx/I’JD = - yav,y/ID. For optimal tuning, the experimental points lie on the second order curve even up to OT = 2.0, and this shows that the damper in this tuning is not sensitive to changes in y. Quantitatively, the relation between theory and experiment would not change appreciably if other average values, or even the zero frequency values, were used. The relative insensitivity of the optimal damping to changes in y is easily demonstrated. For, from equation (1) one has, for a change in w with E = A y/ y,

VIBRATION

DAMPER

WITH SILICONE

OIL

471

Figure 7. Damping values of equilibrated polydimethylsiloxanes. 0, Curve for optimal damping; Q, damping for a Newtonianfluid;0, ~~,x/Z&~,= 0.61; @, ~~,x/ZDr.v= 0.43; 0, &,y/ZDrav = 0.19; ? ,??? , M 10 000; V, M 20 000; 0, 0, M 30 000; A, A, M 50 000; X, Mineral oil. r,, = 1.9 x lo4 N/m’, q,, = 0.811(O).

and in optimal tuning ((1 + rx/lD) = 0) the first order change in dissipated energy is zero. The optimal tuning thus not only gives the highest damping for a given value of x~/(I~o) but the tuning is also stable against changes in the quantity y, a fact that may also be read from Figure 2. This is also important with regard to temperature induced changes in y. The eventual drop in * due to changes in y will be larger, the larger the value of @ is.

5. CONCLUDING REMARKS The experiments show that an increase in dissipated energy by about a factor of two can be obtained, as compared to the maximum dissipated energy in a Newtonian fluid, when silicone oils are used as damping fluids and when the condition for optimal damping is satisfied. Higher values are apparently not possible with the combination of 77(w) and y(w) particular to polydimethylsiloxanes. However these results show a way of improving the performance of dampers, and it is conceivable that other polymers, mixtures or even newly developed polymers can give higher values of y(w)/77 (w), possibly with smaller frequency dependence, and this will allow even higher dissipated energies to be realized. Finally, it should be noted that all the results are immediately applicable to rectilinear vibrations. Equation (1) applies in this case too, if IO is replaced by the mass m of the inertia member and when x is replaced by A/S,where A is the surface at which the fluid is being sheared and S is the clearance. REFERENCES 1. C.

M. HARRIS

McGraw-Hill

and C. E. CREDE (Editors) 1976 Shock and Vibration Handbook. New York: Book Company, second edition.

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AND

J. H. SPURK

2. J. P. DEN HARTOG 1956 Mechanical Vibrations. New York: McGraw-Hill Book Company, fourth edition. 3. J. MICHELE 1976 Rheologica acta 15, 15-22. Die Messung der ersten Normalspannungsdifferenz mit dem “Mechanischen Spektrometer”-das Normalspannungsverhalten von linearen Polysiloxanen und Polyacrylamidbungen. in einer 4. J. H. SPURK 1979 Zngenieur-Archiu 48, 121-127. Der Torsionsschwingungsd&mpfer Fliissigkeit zweiter Ordnung.