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19.3 Representation of Damping in Vibration Analysis

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It is not practical to incorporate detailed microscopic representations of damping in the

dynamic analysis of systems. Instead, simplified models of damping that are representative of

various types of energy dissipation are typically used. Consider a general n-degree-of-freedom

mechanical system. Its motion can be represented by the vector x of n generalized coordinates, xi;

representing the independent motions of the inertia elements. For small displacements, linear

spring elements can be assumed. The corresponding equations of motion may be expressed in the

vector matrix form

Mx€ þ d þ Kx ¼ fðtÞ ð19:27Þ

in which M is the mass (inertia) matrix and K is the stiffness matrix. The forcing-function vector is f ðtÞ:

The damping force vector dðx; x_Þ is generally a nonlinear function of x and x_: The type of damping used

in the system model may be represented by the nature of d that is employed in the system equations.

The various damping models that may be used, as discussed in the previous section, are listed in

Table 19.1. Only the linear viscous damping term given in Table 19.1 is amenable to simplified

mathematical analysis. In simplified dynamic models, other types of damping terms are usually replaced

by an equivalent viscous damping term. Equivalent viscous damping is chosen so that its energy

dissipation per cycle of oscillation is equal to that for the original damping. The resulting equations of

motion are expressed by

Mx€ þ Cx_ þ Kx ¼ f ðtÞ ð19:28Þ

In modal analysis of vibratory systems, the most commonly used model is proportional damping, where

the damping matrix satisfies

C ¼ cmM þ ckK ð19:29Þ

The first term on the right-hand side of Equation 19.29 is known as the inertial damping matrix. The

corresponding damping force on each concentrated mass is proportional to its momentum. It represents

the energy loss associated with a change in momentum (for example, during an impact). The second

term is known as the stiffness damping matrix. The corresponding damping force is proportional to the

rate of change of the local deformation forces at joints near the concentrated mass elements.

Consequently, it represents a simplified form of linear structural damping. If damping is of the

proportional type, it follows that the damped motion can be uncoupled into individual modes. This

means that, if the damping model is of the proportional type, the damped system (as well as the

undamped system) will possess real modes.

TABLE 19.1 Some Common Damping Models Used in Dynamic System

Equations

Damping Type Simplified Model di

Viscous

j cijx_j

Hysteretic

cijx_j

Structural

j cij lxj l sgnðx_jÞ

Structural Coulomb

j cij sgnðx_jÞ

Fluid

j cij lx_j lx_j

Vibration Damping 19-9

19.3.1 Equivalent Viscous Damping

Consider a linear, single-DoF system with viscous damping, subjected to an external excitation.

The equation of motion, for a unit mass, is given by

x€ þ 2zvnx_ þv2

nx ¼ v2

nuðtÞ ð19:30Þ

If the excitation force is harmonic, with frequency v; we have

uðtÞ ¼ u0 cos vt ð19:31Þ

Then, the response of the system at steady state is given by

x ¼ x0 cosðvt þ fÞ ð19:32Þ

in which the response amplitude is

x0 ¼ u0

n 􀀑

ðv2

n 2 v2Þ þ 4z2v2

nv2

􀀜1=2 ð19:33Þ

and the response phase lead is

f ¼ 2tan21 2zvnv

ðv2

n 2 v2Þ ð19:34Þ

The energy dissipation (i.e., damping capacity), DU, per unit mass in one cycle is given by the net work

done by the damping force, fd; thus,

DU ¼

fd dx ¼

ðð2p2fÞv

2f=v

fdx_ dt ð19:35Þ

Since the viscous damping force, normalized with respect to mass (see Equation 19.30), is given by

fd ¼ 2zvnx_ ð19:36Þ

the damping capacity, DUv ; for viscous damping, can be obtained as

DUv ¼ 2zvn

ð2p=v

x_2 dt ð19:37Þ

Finally, using Equation 19.32 in Equation 19.37 we get

DUv ¼ 2px2

0vnvz ð19:38Þ

For any general type of damping (see Table 19.1), the equation of motion becomes

x€ þ dðx; x_Þ þv2

nx ¼ v2

nuðtÞ ð19:39Þ

The energy dissipation in one cycle (Equation 19.35) is given by

DU ¼

ðð2p2fÞ=v

2f=v

dðx; x_Þx_ dt ð19:40Þ

Various damping force expressions, dðx; x_Þ; normalized with respect to mass, are given in Table 19.2.

For fluid damping, for example, the damping capacity is

DUf ¼

ðð2p2fÞ=v

2f=v

clx_lx_2 dt ð19:41Þ

By substituting Equation 19.32 in Equation 19.41 for steady, harmonic motion we obtain

DUf ¼ 83

cx3

0v2 ð19:42Þ

19-10 Vibration and Shock Handbook

By comparing Equation 19.42 with Equation 19.38, the equivalent damping ratio for fluid damping is

obtained as

zf ¼

􀀏 􀀐

x0c ð19:43Þ

in which x0 is the amplitude of steady-state vibrations, as given by Equation 19.33. For the other types of

damping listed in Table 19.1, expressions for the equivalent damping ratio can be obtained in a similar

manner. The corresponding equivalent damping ratio expressions are given in Table 19.2. It should be

noted that, for nonviscous damping types, z is generally a function of the frequency of oscillation, v; and

the amplitude of excitation, u0: It should be noted that the expressions given in Table 19.2 are derived

assuming harmonic excitation. Engineering judgment should be exercised when employing these

expressions for nonharmonic excitations.

For multi-DoF systems that incorporate proportional damping, the equations of motion can be

transformed into a set of one-DoF equations (modal equations) of the type given in Equation 19.30.

In this case, the damping ratio and natural frequency correspond to the respective modal values and,

in particular, v ¼ vn:

19.3.2 Complex Stiffness

Consider a linear spring of stiffness k connected in

parallel with a linear viscous damper of damping

constant c; as shown in Figure 19.5(a). Suppose

that a force, f ; is applied to the system, moving it

through distance x from the relaxed position of the

spring. Then we have

f ¼ kx þ cx_ ð19:44Þ

Suppose that the motion is harmonic, as given by

x ¼ x0 cos vt ð19:45Þ

It is clear that the spring force, kx; is in phase with

the displacement, but the damping force, cx_; has a

908 phase lead with respect to the displacement.

This is because the velocity, x_ ¼ 2x0v sin vt ¼

x0v cosðvt þ p=2Þ; has a 908 phase lead with

respect to x. Specifically, we have

f ¼ kx0 cos vt þ cx0v cos vt þ

􀀏 􀀐

ð19:46Þ

TABLE 19.2 Equivalent Damping Ratio Expressions for Some Common Types of Damping

Damping Type Damping Force, dðx; x_Þ; per Unit Mass Equivalent Damping Ratio, zeq

Viscous 2zvnx_ z

Hysteretic

2vnv

Structural clxl sgnðx_Þ

pvnv

Structural Coulomb c sgnðx_Þ

px0vnv

Fluid clx_lx_

􀀄 v

􀀅

x0 c

f = kx + cx

f = kx + h

w x

x = xo cos w t

(a)

(b)

FIGURE 19.5 Spring element in parallel with (a) a

viscous damper and (b) a hysteretic damper.

Vibration Damping 19-11

This same fact may be represented by using complex numbers, where the in-phase component is

considered as the real part and the 908 phase lead component is considered as the imaginary part with each

component oscillating at the same frequency v: Then, we can write Equation 19.46 in the equivalent form

f ¼ kx þ jvcx ð19:47Þ

This is exactly what we get by starting with the complex representation of the displacement

x ¼ x0 ejvt ð19:48Þ

and substituting it in Equation 19.44. We note that Equation 19.47 may be written as

f ¼ kpx ð19:49Þ

where kp is a “complex” stiffness, given by

kp ¼ k þ jvc ð19:50Þ

Clearly, the system itself and its two components (spring and damper) are real. Their individual forces are

also real. The complex stiffness is simply a mathematical representation of the two force components

(spring force and damping force), which are 908 out of phase, when subjected to harmonic motion. It

follows that the linear damper may be “mathematically” represented by an “imaginary” stiffness. In the case

of viscous damping this imaginary stiffness (and hence, the damping force magnitude) increases linearly

with the frequency, v; of the harmonic motion. The concept of complex stiffness when dealing with

discrete dampers is analogous to the use of complex elastic modulus in material damping, as discussed

earlier in this chapter.

We have noted that, for hysteretic damping, the damping force (or damping stress) is independent of

the frequency in harmonic motion. It follows that a hysteretic damper may be represented by an

equivalent damping constant of

c ¼

v ð19:51Þ

which is valid for a harmonic motion (e.g., modal motion or forced motion) of frequency v: This

situation is shown in Figure 19.5(b). It can be seen that the corresponding complex stiffness is

kp ¼ k þ jh ð19:52Þ

Example 19.3

A flexible system consists of a mass, m; attached to the hysteretic damper and spring combination shown

in Figure 19.5(b). What is the frequency response function of the system relating an excitation force, f ;

applied to the mass and the resulting displacement response, x? Obtain the resonant frequency of the

system. Compare the results with the case for viscous damping.

Solution

For a harmonic motion of frequency v; the equation of motion of the system is

mx€ þ

x_ þ kx ¼ f ð19:53Þ

With a forcing excitation of f ¼ f0 ejvt and the resulting steady-state response, x ¼ x0 ejvt ; where x0 has a

phase difference (i.e., it is a complex function) with respect to f0: Then, in the frequency domain,

substituting the harmonic response x ¼ x0 ejvt into Equation 19.53 we get

2v2m þ

jv þ k

􀀒 􀀓

x ¼ f

19-12 Vibration and Shock Handbook

resulting in the frequency transfer function

f ¼

½k 2 v2m þ jh􀀉 ð19:54Þ

Note that, as usual, this result is obtained simply by substituting jv for d=dt: The magnitude of transfer

function is at a maximum at resonance. This corresponds to a minimum value of

pðvÞ ¼ ðk 2 v2mÞ2 þ h2

If we set dp=dv ¼ 0; we get,

2ðk 2 v2mÞð22vÞ ¼ 0

Hence, the resonant frequency corresponds to the root of

k 2 v2m ¼ 0

This gives the resonant frequency

vr ¼

ffiffiffiffi

ð19:55Þ

Note that, in the case of hysteretic damping, the resonant frequency is equal to the undamped natural

frequency, vn; and, unlike in the case of viscous damping, does not depend on the level of damping itself.

For convenience consider the system response as the spring force

fs ¼ kx ð19:56Þ

Then, a normalized transfer function is obtained, as given by

f ¼ GðjvÞ ¼

1 2 v2 m

k þ j

􀀒 􀀓 ð19:57Þ

or,

f ¼

½1 2 r2 þ ja􀀉 ð19:58Þ

where

r ¼

and a ¼

k ð19:59Þ

which are the normalized frequency and the normalized hysteretic damping, respectively. The magnitude

of the transfer function is

􀀈 􀀈 􀀈 􀀈 􀀈

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 2 r2Þ2 þ a2

p ð19:60Þ

and the phase angle (phase lead) is

/fs=f ¼ 2tan21 a

ð1 2 r2Þ ð19:61Þ

These results are sketched in Figure 19.6.

19.3.3 Loss Factor

We define the damping capacity of a device (damper) as the energy dissipated in a complete cycle of

motion; specifically

DU ¼

fd dx ð19:62Þ

Vibration Damping 19-13

This is given by the area of the hysteresis loop in the displacement force plane. If the initial (total) energy

of the system is denoted by Umax; then the specific damping capacity, D; is given by the ratio

D ¼

Umax ð19:63Þ

The loss factor, h; is the specific damping capacity per radian of the damping cycle. Hence,

h ¼

2pUmax ð19:64Þ

Note that Umax is approximately equal to the maximum kinetic energy and also to the maximum

potential energy of the device when the damping is low.

Equation 19.38 gives the damping capacity per unit mass of a device with viscous damping as

DU ¼ 2px2

0vnvz ð19:65Þ

Here, x0 is the amplitude and v is the frequency of harmonic motion of the device, vn is the undamped

natural frequency and z is the damping ratio. The maximum potential energy per unit mass of the

system is

Umax ¼

0 ¼

nx2

0 ð19:66Þ

Hence, from Equation 19.64, the loss factor for a viscous damped simple oscillator is given by

h ¼

2px2

0vnvz

2p £ 1

2 v2

nx2

0 ¼

2vz

vn ð19:67Þ

For free decay of the system, we have v ¼ vd ø vn; where the latter approximation holds for low

damping. For forced oscillation, the worst response conditions occur when v ¼ vd ø vn; which is

what one must consider with regard to energy dissipation. In either case, the loss factor is approximately

−

FIGURE 19.6 Frequency transfer function of a simple oscillator with hysteretic damping.

19-14 Vibration and Shock Handbook

given by

h ¼ 2z ð19:68Þ

For other types of damping, Equation 19.68 will still hold when the equivalent damping ratio, zeq;

(see Table 19.2) is used in place of z:

The loss factors of some common materials are given in Table 19.3. Definitions of useful damping

parameters, as defined here, are summarized in Table 19.4. Expressions of loss factors for some useful

damping models are given in Table 19.5.

TABLE 19.3 Loss Factors of Some Useful Materials

Material Loss Factor h ø 2z

Aluminum 2 £ 1025 to 2 £ 1023

Concrete 0.02 to 0.06

Glass 0.001 to 0.002

Rubber 0.1 to 1.0

Steel 0.002 to 0.01

Wood 0.005 to 0.01

TABLE 19.4 Definitions of Damping Parameters

Parameter Definition Mathematical Formula

Damping capacity ðDUÞ Energy dissipated per cycle

of motion (area of

displacement – force hysteresis loop)

fd dx

Damping capacity per

volume ðdÞ

Energy dissipated per cycle

per unit material volume

(area of strain – stress hysteresis loop)

s d1

Specific damping

capacity ðDÞ

Ratio of energy dissipated

per cycle ðDUÞ to the initial maximum

energy ðUmax Þ Note: for low damping,

Umax ¼ maximum potential energy

¼ maximum kinetic energy

Umax

Loss factor ðhÞ Specific damping capacity per

unit angle of cycle.

Note: for low damping,

h ¼ 2 £ damping ratio

2pUmax

TABLE 19.5 Loss Factors for Several Material Damping Models

Material Damping Model Stress – Strain Constitute Relation Loss Factor ðhÞ

Viscoelastic Kelvin – Voigt s ¼ E1 þ Ep d1

vEp

Hysteretic Kelvin – Voigt s ¼ E1 þ

Viscoelastic standard linear solid s þ cs

dt ¼ E1 þ Ep d1

vEp

ð1 2 cs E=Ep Þ

ð1 þ v2 cs Þ

Hysteretic standard linear solid s þ cs

dt ¼ E1 þ

ð1 2vcsE=E~ Þ

ð1 þ v2 cs Þ

Vibration Damping 19-15

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