19.5 Interface Damping

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In many practical applications damping is generated at the interface of two sliding surfaces. This is the

case, for example, in bearings, gears, screws, and guideways. Even though this type of damping is

commonly treated under structural damping, due to its significance we will consider it again here in

more detail, as a category of its own.

Interface damping was formally considered by DaVinci in the early 1500s and again by Coulomb in the

1700s. The simplified model used by them is the well-known Coulomb friction model as given by

f ¼ mR sgnðvÞ ð19:104Þ

where

f ¼ the frictional force that opposes the motion

R ¼ the normal reaction force between the sliding surfaces

v ¼ the relative velocity between the sliding surfaces

m ¼ the coefficient of friction

Note that the signum function “sgn” is used to emphasize that f is in the opposite direction of v:

This simple model is not expected to provide accurate results in all cases of interface damping. It is

known that, apart from the loading conditions, interface damping depends on a variety of factors such as

material properties, surface characteristics, nature of lubrication, geometry of the moving parts, and the

magnitude of the relative velocity.

0.10

0.05

0.01

0

0.001 0.01 0.1 1.0 10.0

Amplitude of Motion (cm)

Damping Ratio z

FIGURE 19.14 Effect of vibration amplitude on damping in structures.

TABLE 19.6 Typical Damping Values Suggested by ASME for Seismic Applications

System Damping Ratio ðz%Þ

OBE SSE

Equipment and large diameter piping systems (. 12 in. diameter) 2 3

Small diameter piping systems (# 12 in. diameter) 1 2

Welded steel structures 2 4

Bolted steel structures 4 7

Prestressed concrete structures 2 5

Reinforced concrete structures 4 7

19-26 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

A somewhat more complete model for interface damping, incorporating the following characteristics,

is shown in Figure 19.15:

1. Static and dynamic friction, with stiction and stick – slip behavior.

2. Conventional Coulomb friction (Region 1).

3. A drop in dynamic friction, with a negative slope, before increasing again. This is known as the

“Stribeck effect” (Region 2).

4. Conventional viscous damping (Region 3).

These characteristics cover the behavior of interface damping that is commonly observed in practice.

In particular, suppose that a force is exerted to generate a relative motion between two surfaces. For small

values of the force, there will not be a relative motion, in view of friction. The minimum force, fs; that is

needed for the motion to start is the static frictional force. The force that is needed to maintain the

motion will drop instantaneously to fd; as the motion begins. It is as though initially the two surfaces were

“stuck” and fs is the necessary breakaway force. Hence, this characteristic is known as stiction. The

minimum force fd that is needed to maintain the relative motion between the two surfaces is called

dynamic friction. In fact, under dynamic conditions, it is possible for “stick – slip” to occur where

repeated sticking and breaking away cycles of intermittent motion take place. Clearly, such “chattering”

motion corresponds to instability (for example, in machine tools). It is an undesirable effect and should

be avoided.

After the relative motion begins, conventional Coulomb type damping behavior may dominate for

small relative velocities, as represented in Region 1. For lubricated surfaces, at low relative velocities, there

will be some solid-to-solid contact that generates a Coulomb-type damping force. As the relative speed

increases, the degree of this solid-to-solid contact will decrease and the damping force will drop, as in

Region 2 of Figure 19.15. This characteristic is known as the Stribeck effect. Since the slope of the friction

curve is negative in Regions 1 and 2, this corresponds to the unstable region. As the relative velocity is

further increased, in fully lubricated surfaces, viscous-type damping will dominate, as shown in Region 3

of Figure 19.15. This is the stable region. It follows that a combined model of interface damping may be

expressed as

f ¼

fs for v ¼ 0

fsbðvÞsgnðvÞ þ bv for v – 0

(

ð19:105Þ

Damping

Force f

fs

fd

Region

1

Region

2

Region

3

Unstable Stable

Relative Velocity v

FIGURE 19.15 Main characteristics of interface damping.

Vibration Damping 19-27

© 2005 by Taylor & Francis Group, LLC

Note that fsbðvÞ is a nonlinear function of velocity that will represent both dynamic friction (for v . 0)

and the Stribeck effect. Models that have been used to represent this effect include the following

fsb ¼

fd

1 þ ðv=vcÞ2 ð19:106Þ

fsb ¼ fd e2ðv=vc Þ2

ð19:107Þ

and

fsb ¼ ð fd þ alvl1=2ÞsgnðvÞ ð19:108Þ

Note that fd represents dynamic Coulomb friction and vc and a are modal parameters.

Example 19.5

An object of mass m rests on a horizontal

surface and is attached to a spring of stiffness k;

as shown in Figure 19.16. The mass is pulled so

that the extension of the spring is x0; and is

moved from rest from that position. Determine

the subsequent sliding motion of the object. The

coefficient of friction between the object at the

horizontal surface is m:

Solution

Note that, when the object moves to the left, the frictional force, mmg acts to the right, and vice versa.

Now, consider the first cycle of motion, stating from rest with x ¼ x0 moving to the left, coming to rest

with the spring compressed and then moving to the right.

First Half Cycle (Moving to Left)

The equation of motion is

mx€ ¼ 2kx þmmg ðiÞ

or

x€ þv2

nx ¼ mg ðiiÞ

where vn ¼

ffiffiffiffiffi

k=m p is the undamped material frequency. Equation ii has a homogeneous solution of

xh ¼ A1 sinðvntÞ þ A2 cosðvntÞ ðiiiÞ

and a particular solution of

xp ¼

mg

v2

n ðivÞ

Hence, the total solution is

x ¼ A1 sinðvntÞ þ A2 cosðvntÞ þ

mg

v2

n ðvÞ

Using the initial conditions x ¼ x0 and x_ ¼ 0 at t ¼ 0; we get A1 ¼ 0 and A2 ¼ x0 2 ðmg=v2

nÞ Hence,

Equation v becomes

x ¼ x0 2

mg

v2

n

􀀏 􀀐

cosðvntÞ þ

mg

v2

n ðviÞ

At the end of this half cycle we have x_ ¼ 0 or sin vnt ¼ 0: Hence the corresponding

time is t ¼ p=vn: Substituting this in Equation vi, the corresponding position of the object

k

x

mg

m

FIGURE 19.16 An object sliding against Coulomb

friction.

19-28 Vibration and Shock Handbook

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is (note: cos p ¼ 21)

xl1 ¼ 2 x0 2

2mg

v2

n

􀀏 􀀐

ðviiÞ

Second Half Cycle (Moving to Right)

The equation of motion is

mx€ ¼ 2kx 2mmg ðviiiÞ

or

x€ þv2

n x ¼ 2mg ðixÞ

The corresponding response is given by

x ¼ B1 sinðvntÞ þ B2 cosðvntÞ 2

mg

v2

n ðxÞ

Using the initial conditions x ¼ 2ðx0 2 ð2mg=v2

nÞÞ and x_ ¼ 0 at t ¼ p=vn; we get B1 ¼ 0 and B2 ¼

x0 2 ð3mg=v2

nÞ: Hence, Equation x becomes

x ¼ x0 2

3mg

v2

n

􀀏 􀀐

cosðvntÞ 2

mg

v2

n ðxiÞ

The object will come to rest ðx_ ¼ 0Þ next at t ¼ 2p=vn; hence the position of the object at the end of the

present half cycle would be

x1 ¼ x0 2

4mg

v2

n ðxiiÞ

The response for the next cycle is determined by substituting x1 as given by Equation xii with Equation vi

for the left motion and with Equation xi for the right motion. Then, we can express the general response

as

left motion in cycle i : x ¼ ½x0 2 ð4i 2 3ÞD􀀉 cos vnt þ D ðxiiiÞ

right motion in cycle i : x ¼ ½x0 2 ð4i 2 1ÞD􀀉 cos vnt 2 D ðxivÞ

where

D ¼

mg

v2

n ðxvÞ

Note that the amplitude of the harmonic part of the response should be positive for that half cycle of

motion to be possible. Hence, we must have

x0 . ð4i 2 3ÞD for left motion in cycle i

x0 . ð4i 2 1ÞD for right motion in cycle i

Also note from Equation xiii and Equation xiv that the equilibrium position for the left motion is þD

and for the right motion is 2D: A typical response curve is sketched in Figure 19.17.

19.5.1 Friction in Rotational Interfaces

Friction in gear transmissions, rotary bearings, and other rotary joints has a somewhat similar behavior.

Of course, the friction characteristics will depend on the nature of the devices and also the loading

conditions, but, experiments have shown that the frictional behavior of these devices may be

represented by the interface damping model given here. Typically, experimental results are presented as

Vibration Damping 19-29

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curves of coefficient of friction (frictional force/

normal force) vs. relative velocity of the two

sliding surfaces. In the case of rotary bearings, the

rotational speed of the shaft is used as the relative

velocity, while for gears; the pitch line velocity is

used. Experimental results for a pair of spur gears

are shown in Figure 19.18.

What is interesting to notice from the result is

the fact that, for this type of rotational device, the

damping behavior may be approximated by two

straight line segments in the velocity – friction

plane; the first segment having a sharp negative

slope and the second segment having a

moderate positive slope that represents the equivalent viscous damping constant, as shown in

Figure 19.19.

19.5.2 Instability

Unstable behavior or self-excited vibrations, such as stick – slip and chatter, that are exhibited by

interacting devices such as metal removing tools (e.g., lathes, drills, and milling machines) may be easily

Position

x

xo

D

−D

0

p/wn 2p/wn Time t

FIGURE 19.17 A typical cyclic response under Coulomb friction.

Coefficient of Friction

0.05

0.04

0.03

0.02

0.01

0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Pitch Line Velocity (m /s)

FIGURE 19.18 Frictional characteristics of a pair of spur gears.

Relative Velocity

Coefficient of Friction

FIGURE 19.19 A friction model for rotatory devices.

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explained using the interface damping model. In particular, it is noted that the model has a region

of negative slope (or negative damping constant), which corresponds to low relative velocities, and a

region of positive slope, which corresponds to high relative velocities. Consider the single-DoF model:

mx€ þ bx_ þ kx ¼ 0 ð19:109Þ

without an external excitation force. Initially the velocity is x_ ¼ 0: But, in this region, the damping

constant, b; will be negative and hence the system will be unstable. Hence, a slight disturbance will result

in a steadily increasing response. Subsequently, x_ will increase above the critical velocity where b will be

positive and the system will be stable. As a result, the response will steadily decrease. This growing and

decaying cycle will be repeated at a frequency that primarily depends on the inertia and stiffness

parameters ðm and kÞ of the system. Chatter is caused in this manner in interfaced devices.

Bibliography

Blevins, R.D. 1977. Flow-Induced Vibration, Van Nostrand Reinhold, New York.

den Hartog, J.P. 1956. Mechanical Vibrations, McGraw-Hill, New York.

de Silva, C.W., Dynamic beam model with internal damping, rotatory inertia and shear deformation,

AIAA J., 14, 5, 676 – 680, 1976.

de Silva, C.W., Optimal estimation of the response of internally damped beams to random loads in the

presence of measurement noise, J. Sound Vib., 47, 4, 485 – 493, 1976.

de Silva, C.W., An algorithm for the optimal design of passive vibration controllers for flexible systems,

J. Sound Vib., 74, 4, 495 – 502, 1982.

de Silva, C.W. 2000. VIBRATION — Fundamentals and Practice, CRC Press, Boca Raton, FL.

de Silva, C.W. 2004. MECHATRONICS — An Integrated Approach, CRC Press, Boca Raton, FL.

Ewins, D.J. 1984. Modal Testing: Theory and Practice. Research Studies Press Ltd, Letchworth, England.

Inman, D.J. 1996. Engineering Vibration, Prentice Hall, Englewood Cliffs, NJ.

Irwin, J.D. and Graf, E.R. 1979. Industrial Noise and Vibration Control, Prentice Hall, Englewood

Cliffs, NJ.

Van de Vegte, J. and de Silva, C.W., Design of passive vibration controls for internally damped beams by

modal control techniques, J. Sound Vib., 45, 3, 417 – 425, 1976.

Vibration Damping 19-31

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