21.9 Viscous Damping — Need for Caution

Back

Recent experiments have shown important subtleties of viscous damping (Peters, 2003a, 2003b, 2003c,

2003d). It is true that the dissipation at a specified frequency can be adequately modeled by simply

multiplying the velocity term in the differential equation by a coefficient. It is not proper, however, to call

FIGURE 21.34 Same as Figure 21.33, but after the pendulum had stabilized.

Experimental Techniques in Damping 21-29

© 2005 by Taylor & Francis Group, LLC

this coefficient a constant, since the damping coefficient is frequency dependent and also involves the

density as well as the viscosity of the fluid in which oscillation takes place.

Some engineers have known about the history term, which is most simply treated in the case of a

sphere executing simple harmonic motion. The friction force acting on the sphere in this case can be

reasonably approximated by

fharmonic ¼ 6pha 1 þ CH

a

d

􀀏 􀀐

v; d ¼

ffiffiffiffiffi

2h

vr

s

; ðCH ! 1 as v ! 0Þ ð21:11Þ

where v is the angular frequency of oscillation, a is the radius of the sphere; and for the fluid, h and r are

its viscosity and density, respectively. Only in the limit of zero frequency does the damping reduce to the

form that one expects on the basis of Stokes’ law of viscous friction (steady flow).

Using Equation 21.11 in the equation of motion for a pendulum yields for the Q

Qv ¼

Iv

6pha 1 þ

a

d

􀀏 􀀐

L2

; d ¼

ffiffiffiffiffi

2h

vr

s

ð21:12Þ

where I is the moment of inertia, and L is the distance from the axis to the center of the sphere. Typically,

the ratio a=d is significantly greater than unity so that the damping is governed by the surface area of the

sphere rather than by its radius. These complexities of viscous damping are summarized in Box 21.3.

Reasonable experimental validation of the estimate for Q was provided, as demonstrated in

Figure 21.35.

The instrument in this case was a compound pendulum in which a mass was located above the

axis of rotation, as well as the usual situation of mass below the axis. The water damping was

provided through a small sphere at the bottom of the pendulum, immersed in water held by a

rectangular container.

If the history term in Equation 21.12 is ignored there can be huge errors. For example, in the case of

water damping, the damping can be underestimated by 1000 to 3000%, as shown in Figure 21.36.

At low frequencies, it is also important to correct for the influence of hysteretic damping of the

pendulum. Figure 21.37 shows the large errors that occur when one fails to do so.

For some cases, buoyancy and added mass of the fluid are also quite significant to the frequency of

oscillation, as shown in Figure 21.38.

Box 21.3

COMPLEXITIES OF VISCOUS DAMPING

Friction force is not a function only of viscosity h; it also depends on density r and angular

frequency v

fharmonic ¼ 6pha 1 þ CH

a

d

􀀏 􀀐

v; d ¼

ffiffiffiffiffi

2h

vr

s

; ðCH ! 1 as v ! 0Þ

resulting in a complicated frequency dependence for the Q of viscous damping

Qv ¼

Iv

6pha 1 þ

a

d

􀀏 􀀐

L2

; d ¼

ffiffiffiffiffi

2h

vr

s

21-30 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Навигация