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19.2 Types of Damping

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There is some form of mechanical energy dissipation in any dynamic system. In the modeling of systems,

damping can be neglected if the mechanical energy that is dissipated during the time duration of interest

is small in comparison to the initial total mechanical energy of excitation in the system. Even for highly

damped systems, it is useful to perform an analysis with the damping terms neglected, in order to study

several crucial dynamic characteristics, e.g., modal characteristics (undamped natural frequencies and

mode shapes).

Several types of damping are inherently present in a mechanical system. If the level of damping that is

available in this manner is not adequate for proper functioning of the system then external damping devices

may be added either during the original design or during subsequent design modifications of the system.

Three primary mechanisms of damping are important in the study of mechanical systems. They are:

1. Internal damping (of material)

2. Structural damping (at joints and interfaces)

3. Fluid damping (through fluid – structure interactions)

Internal (material) damping results from mechanical energy dissipation within the material due to

various microscopic and macroscopic processes. Structural damping is caused by mechanical energy

dissipation resulting from relative motions between components in a mechanical structure that has

common points of contact, joints or supports. Fluid damping arises from the mechanical energy

dissipation resulting from drag forces and associated dynamic interactions when a mechanical system or

its components move in a fluid.

Two general types of external dampers may be added to a mechanical system in order to improve its

energy dissipation characteristics. They are:

1. Passive dampers

2. Active dampers

Passive dampers are devices that dissipate energy through some kind of motion, without needing an

external power source or actuators. Active dampers have actuators that need external sources of power.

They operate by actively controlling the motion of the system that needs damping. Dampers may be

considered as vibration controllers. In the present chapter, the emphasis will be on damping that is

inherently present in a mechanical system.

19.2.1 Material (Internal) Damping

Internal damping of materials originates from the energy dissipation associated with microstructure

defects, such as grain boundaries and impurities; thermoelastic effects caused by local temperature

gradients resulting from nonuniform stresses, as in vibrating beams; eddy current effects in ferromagnetic

19-2 Vibration and Shock Handbook

materials; dislocation motion in metals; and chain

motion in polymers. Several models have been

employed to represent energy dissipation caused

by internal damping. This variety of models is

primarily a result of the vast range of engineering

materials; no single model can satisfactorily

represent the internal damping characteristics of

all materials. Nevertheless, two general types of

internal damping can be identified: viscoelastic

damping and hysteretic damping. The latter term

is actually a misnomer, because all types of

internal damping are associated with hysteresis

loop effects. The stress ðsÞ and strain ð1Þ relations

at a point in a vibrating continuum possess a

hysteresis loop, such as the one shown in

Figure 19.1. The area of the hysteresis loop gives

the energy dissipation per unit volume of the

material, per stress cycle. This is termed the perunit-

volume damping capacity, and is denoted by d: It is clear that d is given by the cyclic integral

d ¼

s d1 ð19:1Þ

In fact, for any damped device, there is a corresponding hysteresis loop in the displacement – force

plane as well. In this case, the cyclic integral of force with respect to the displacement, which is the area

of the hysteresis loop, is equal to the work done against the damping force. It follows that this integral

(loop area) is the energy dissipated per cycle of motion. This is the damping capacity which, when

divided by the material volume, gives the per-unit-volume damping capacity as before.

It should be clear that, unlike a pure elastic force (e.g., a spring force), a damping force cannot be a

function of displacement ðqÞ alone. The reason is straightforward. Consider a force f ðqÞ which depends

on q alone. Then, for a particular displacement point, q; of the component the force will be the same

regardless of the direction of motion (i.e., the sign of q_ ). It follows that, in a loading and unloading cycle,

the same path will be followed in both directions of motion. Hence, a hysteresis loop will not be formed.

In other words, the net work done in a complete cycle of motion will be zero. Next consider a force f ðq; q_)

which depends on both q and q_: Then, at a given displacement point, q; the force will depend on q_ as well.

Hence, force in one direction of motion will be different from that in the opposite direction. As a result, a

hysteresis loop will be formed, which corresponds to work done against the damping force (i.e., energy

dissipation).We can conclude then that the damping force has to depend on a relative velocity, q_; in some

manner. In particular, Coulomb friction, which does not depend on the magnitude of q_; does depend on

the sign (direction) of q_:

19.2.1.1 Viscoelastic Damping

For a linear viscoelastic material, the stress – strain relationship is given by a linear differential equation

with respect to time, having constant coefficients. A commonly employed relationship is

s ¼ E1 þ Ep d1

dt ð19:2Þ

which is known as the Kelvin – Voigt model. In Equation 19.2, E is Young’s modulus and Ep is a

viscoelastic parameter that is assumed to be time independent. The elastic term E1 does not contribute

to damping, and, as noted before, mathematically, its cyclic integral vanishes. Consequently, for the

Kelvin – Voigt model, damping capacity per unit volume is

dv ¼ Ep

þ d1

d1p ð19:3Þ

Stress

Strain e

Area = Damping Capacity

per Unit Volume

smax

emax

FIGURE 19.1 A typical hysteresis loop for mechanical

damping.

Vibration Damping 19-3

For a material that is subjected to a harmonic (sinusoidal) excitation, at steady state, we have

1 ¼ 1max cos vt ð19:4Þ

When Equation 19.4 is substituted in Equation 19.3, we obtain

dv ¼ pvEp12

max ð19:5Þ

Now, 1 ¼ 1max when t ¼ 0 in Equation 19.4, or when d1=dt ¼ 0: The corresponding stress, according to

Equation 19.2, is smax ¼ E1max: It follows that

dv ¼

pvEps 2

max

E2 ð19:6Þ

These expressions for dv depend on the frequency of excitation, v:

Apart from the Kelvin – Voigt model, two other models of viscoelastic damping are also commonly

used. They are, the Maxwell model given by

s þ cs

dt ¼ Ep d1

dt ð19:7Þ

and the standard linear solid model given by

s þ cs

dt ¼ E1 þ Ep d1

dt ð19:8Þ

It is clear that the standard linear solid model represents a combination of the Kelvin – Voigt model

and the Maxwell model, and is the most accurate of the three. But, for most practical purposes, the

Kelvin – Voigt model is adequate.

19.2.1.2 Hysteretic Damping

It was noted above that the stress, and hence the internal damping force, of a viscoelastic damping

material depends on the frequency of variation of the strain (and consequently the frequency of motion).

For some types of material, it has been observed that the damping force does not significantly depend on

the frequency of oscillation of strain (or frequency of harmonic motion). This type of internal damping is

known as hysteretic damping.

Damping capacity per unit volume ðdhÞ for hysteretic damping is also independent of the frequency of

motion and can be represented by

dh ¼ Js n

max ð19:9Þ

A simple model that satisfies Equation 19.9, for the case of n ¼ 2; is given by

s ¼ E1 þ

dt ð19:10Þ

which is equivalent to using a viscoelastic parameter, Ep ; that depends on the frequency of motion in

Equation 19.2 according to Ep ¼ E~ =v:

Consider the case of harmonic motion at frequency v; with the material strain given by

1 ¼ 10 cos vt ð19:11Þ

Then, Equation 19.10 becomes

s ¼ E10 cos vt 2 E~10 sin vt ¼ E1 cos vt þ E~10 cos vt þ

􀀏 􀀐

ð19:12Þ

Note that the material stress has two components, as given by the right-hand side of Equation 19.12. The

first component corresponds to the linear elastic behavior of a material and is in phase with the strain.

The second component of stress, which corresponds to hysteretic damping, is 908 out of phase. (This

stress component leads the strain by 908.) A convenient mathematical representation is possible, by using

the usual complex form of the response according to

1 ¼ 10 ejvt ð19:13Þ

19-4 Vibration and Shock Handbook

Then, Equation 19.10 becomes

s ¼ ðE þ jE~ Þ1 ð19:14Þ

It follows that this form of simplified hysteretic damping may be represented by using a complex

modulus of elasticity, consisting of a real part which corresponds to the usual linear elastic (energy

storage) modulus (or Young’s modulus) and an imaginary part which corresponds to the hysteretic loss

(energy dissipation) modulus.

By combining Equation 19.2 and Equation 19.10, a simple model for combined viscoelastic and

hysteretic damping may be given by

s ¼ E1 þ Ep þ

􀁻 !

dt ð19:15Þ

The equation of motion for a system whose damping is represented by Equation 19.15 can be deduced

from the pure elastic equation of motion by simply substituting E by the operator

E þ Ep þ

􀁻 !

›

›t

in the time domain.

Example 19.1

Determine the equation of flexural motion of a nonuniform slender beam whose material has both

viscoelastic and hysteretic damping.

Solution

The Bernoulli – Euler equation of bending motion on an undamped beam subjected to a dynamic load of

f ðx; tÞ per unit length, is given by

›2

›x2 EI

›2q

›x2 þ rA

›2q

›t2 ¼ f ðx; tÞ ð19:16Þ

Here, q is the transverse motion at a distance, x; along the beam. Then, for a beam with material damping

(both viscoelastic and hysteretic) we can write

›2

›x2 EI

›2q

›x2 þ

›2

›x2 Ep þ

􀁻 !

›3q

›t ›x2 þ rA

›2q

›t2 ¼ f ðx; tÞ ð19:17Þ

in which v is the frequency of the external excitation f ðx; tÞ in the case of steady forced vibrations. In the

case of free vibration, however, v represents the frequency of free vibration decay. Consequently, when

analyzing the modal decay of free vibrations, v in Equation 19.17 should be replaced by the appropriate

frequency ðviÞ of modal vibration in each modal equation. Hence, the resulting damped vibratory system

possesses the same normal mode shapes as the undamped system. The analysis of the damped case is very

similar to that for the undamped system.

19.2.2 Structural Damping

Structural damping is a result of mechanical energy dissipation caused by friction due to the relative

motion between components and by impacting or intermittent contact at the joints in a mechanical

system or structure. Energy dissipation behavior depends on the details of the particular mechanical

system. Consequently, it is extremely difficult to develop a generalized analytical model that would

satisfactorily describe structural damping. Energy dissipation caused by rubbing is usually represented by

a Coulomb friction model. Energy dissipation caused by impacting, however, should be determined from

the coefficient of restitution of the two members that are in contact.

The most common method of estimating structural damping is by measurement. The measured

values, however, represent the overall damping in the mechanical system. The structural damping

component is obtained by subtracting the values corresponding to other types of damping, such as

material damping, present in the system (estimated by environment-controlled experiments, previous

data, and so forth) from the overall damping value.

Vibration Damping 19-5

Usually, internal damping is negligible compared

to structural damping. A large proportion of

mechanical energy dissipation in tall buildings,

bridges, vehicle guideways, and many other civil

engineering structures and in machinery, such as

robots and vehicles, takes place through the

structural damping mechanism. A major form of

structural damping is the slip damping that results

from energy dissipation by interface shear at a

structural joint. The degree of slip damping that is

directly caused by Coulomb (dry) friction depends

on such factors as joint forces (for example, bolt

tensions), surface properties and the nature of the

materials of the mating surfaces. This is associated

with wear, corrosion, and general deterioration of

the structural joint. In this sense, slip damping is

time-dependent. It is a common practice to place

damping layers at joints to reduce undesirable

deterioration of the joints. Sliding causes shear

distortions in the damping layers, causing energy

dissipation by material damping and also through

Coulomb friction. In this way, a high level of

equivalent structural damping can be maintained

without causing excessive joint deterioration.

These damping layers should have a high stiffness

(as well as a high specific-damping capacity) in

order to take the structural loads at the joint.

For structural damping at a joint, the damping

force varies as slip occurs at the joint. This is

primarily caused by local deformations at the joint,

which occur with slipping. A typical hysteresis

loop for this case is shown in Figure 19.2(a). The

arrows on the hysteresis loop indicate the direction

of relative velocity. For idealized Coulomb friction,

the frictional force ðFÞ remains constant in each direction of relative motion. An idealized hysteresis loop

for structural Coulomb damping is shown in Figure 19.2(b). The corresponding constitutive relation is

f ¼ c sgnðq_Þ ð19:18Þ

in which f is the damping force, q is the relative displacement at the joint and c is a friction parameter.

A simplified model for structural damping caused by local deformation may be given by

f ¼ clqlsgnðq_Þ ð19:19Þ

The corresponding hysteresis loop is shown in Figure 19.2(c). Note that the signum function is defined by

sgnðvÞ ¼

1 for v $ 0

21 for v , 0

(

ð19:20Þ

19.2.3 Fluid Damping

Consider a mechanical component moving in a fluid medium. The direction of relative motion is shown

parallel to the y-axis in Figure 19.3. Local displacement of the element relative to the surrounding fluid is

denoted by qðx; y; tÞ:

Force

(Displacement)

(a)

Force

(Displacement)

(b)

Force

(Displacement)

(c)

FIGURE 19.2 Some representative hysteresis loops:

(a) typical structural damping; (b) Coulomb friction

model; and (c) simplified structural damping model.

19-6 Vibration and Shock Handbook

The resulting drag force per unit area of

projection on the x – z plane is denoted by fd:

This resistance is the cause of mechanical energy

dissipation in fluid damping. It is usually

expressed as

fd ¼ 12

cdrq_2 sgnðq_Þ ð19:21Þ

in which q_ ¼ ›qðx; z; tÞ=›t is the relative velocity.

The drag coefficient, cd; is a function of the

Reynold’s number and the geometry of the

structural cross section. A net damping effect is

generated by viscous drag produced by the boundary layer effects at the fluid – structure interface, and by

pressure drag produced by the turbulent effects resulting from flow separation at the wake. The two

effects are illustrated in Figure 19.4. Fluid density is r: For fluid damping, the damping capacity per unit

volume associated with the configuration shown in Figure 19.3 is given by

df ¼

ÞðLx

ðLz

fd dz dx dqðx; z; tÞ

Lx Lz q0 ð19:22Þ

in which, Lx and Lz are cross-sectional dimensions of the element in the x and y-directions, respectively,

and q0 is a normalizing amplitude parameter for relative displacement.

Example 19.2

Consider a beam of length L and uniform rectangular cross section that is undergoing transverse

vibration in a stationary fluid. Determine an expression for the damping capacity per unit volume for

this fluid – structure interaction.

Solution

Suppose that the beam axis is along the x-direction and the transverse motion is in the z-direction. There is

no variation in the y-direction, and hence, the length parameters in this direction cancel out.

df ¼

ÞðL

fd dx dqðx; tÞ

Lq0

df ¼

ðT

ðL

fdq_ðx; tÞdx dt

Lq0 ð19:23Þ

FIGURE 19.3 A body moving in a fluid medium.

FIGURE 19.4 Mechanics of fluid damping.

Vibration Damping 19-7

in which T is the period of the oscillations. Assuming constant cd; we substitute Equation 19.21

into Equation 19.23:

df ¼

cdr

Lq0

ðL

ðT

lq_l3dt dx ð19:24Þ

For steady-excited harmonic vibration at frequency v and shape function QðxÞ (or for free-modal

vibration at natural frequency v and mode shape QðxÞ) we have

qðx; tÞ ¼ qmaxQðxÞ sin vt ð19:25Þ

In this case, with the change of variable u ¼ vt; Equation 19.24 becomes

df ¼ 2cdr

max

Lq0

ðL

lQðxÞl3dx v2

ðp=2

cos3u du

df ¼

cdrq3

maxv2

ðL

lQðxÞl3dx

Lq0

Note. The integration interval of t ¼ 0 to T becomes u ¼ 0 to 2p or four times that from u ¼ 0 to p=2:

If the normalizing parameter is defined as

q0 ¼

qmax

ðL

lQðxÞl3 dx

then, we get

df ¼

cdrq2

maxv2 ð19:26Þ

A useful classification of damping is given in Box 19.1.

Box 19.1

DAMPING CLASSIFICATION

Type of Damping Origin Typical Constitutive Relation

Internal damping Material properties Viscoelastic

s ¼ E1 þ Ep d1

Hysteretic

s ¼ E1 þ

Structural damping Structural joints and interfaces Structural deformation

f ¼ clql sgnðq_Þ

Coulomb f ¼ c sgnðq_Þ

General interface

f ¼

fs for v ¼ 0

fsbðvÞ sgnðvÞ for v – 0

(

Fluid damping Fluid – structure interactions fd ¼ 12

cdrq_2 sgnðq_Þ

19-8 Vibration and Shock Handbook

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