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19.4 Measurement of Damping

Back

Damping may be represented by various parameters (such as specific damping capacity, loss factor,

Q-factor, and damping ratio) and models (such as viscous, hysteretic, structural, and fluid). Before

attempting to measure damping in a system, we need to decide on a representation (model) that will

adequately characterize the nature of mechanical energy dissipation in the system. Next, we should

decide on the parameter or parameters of the model that need to be measured.

It is extremely difficult to develop a realistic yet tractable model for damping in a complex piece of

equipment operating under various conditions of mechanical interaction. Even if a satisfactory damping

modal is developed, experimental determination of its parameters could be tedious. A major difficulty

arises because it usually is not possible to isolate various types of damping (for example, material,

structural, and fluid) from an overall measurement. Furthermore, damping measurements must be

conducted under actual operating conditions for them to be realistic.

If one type of damping (say, fluid damping) is eliminated during the actual measurement then it would

not represent the true operating conditions. This would also eliminate possible interacting effects of the

eliminated damping type with the other types. In particular, overall damping in a system is not generally

equal to the sum of the individual damping values when they are acting independently. Another

limitation of computing equivalent damping values using experimental data arises because it is assumed

for analytical simplicity that the dynamic system behavior is linear. If the system is highly nonlinear, a

significant error could be introduced into the damping estimate. Nevertheless, it is customary to assume

linear viscous behavior when estimating damping parameters using experimental data.

There are two general ways by which damping measurements can be made: using a time – response

record and using a frequency – response function of the system to estimate damping.

19.4.1 Logarithmic Decrement Method

This is perhaps the most popular time – response method that is used to measure damping. When

a single-DoF oscillatory system with viscous damping (see Equation 19.30) is excited by an impulse

input (or an initial condition excitation), its response takes the form of a time decay (see Figure 19.7),

given by

yðtÞ ¼ y0 expð2zvntÞ sin vdt ð19:69Þ

in which the damped natural frequency is given by

vd ¼

ffiffiffiffiffiffiffiffi

1 2 z2

vn ð19:70Þ

Time t

Ai+r

Displacement

y (t)

FIGURE 19.7 Impulse response of a simple oscillator.

19-16 Vibration and Shock Handbook

If the response at t ¼ ti is denoted by yi; and the response at t ¼ ti þ 2pr=vd is denoted by yiþr ; then,

from Equation 19.69, we have

yiþr

yi ¼ exp 2z

2pr

􀀏 􀀐

; i ¼ 1; 2; …; n

In particular, suppose that yi corresponds to a peak point in the time decay function, having magnitude

Ai; and that yiþr corresponds to the peak point r cycles later in the time history, and its magnitude is

denoted by Aiþr (see Figure 19.7). Even though the above equation holds for any pair of points that are

r periods apart in the time history, the peak points seem to be the appropriate choice for measurement

in the present procedure, as these values would be more prominent than any arbitrary points in a

response – time history. Then,

Aiþr

Ai ¼ exp 2z

2pr

􀀏 􀀐

¼ exp 2

z ffiffiffiffiffiffiffiffi

1 2 z2

p 2pr

" #

where Equation 19.70 has been used. Then, the logarithmic decrement d is given by (per unit cycle)

d ¼

Aiþr

􀀏 􀀐

2pz ffiffiffiffiffiffiffiffi

1 2 z2

p ð19:71Þ

or the damping ratio may be expressed as

z ¼

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ ð2p=dÞ2

p ð19:72Þ

For low damping (typically, z , 0:1), vd ø vn and Equation 19.71 become

Aiþr

ø expð2z2prÞ ð19:73Þ

z ¼

2pr

Aiþr

􀀏 􀀐

for z , 0:1 ð19:74Þ

This is in fact the “per-radian” logarithmic decrement.

The damping ratio can be estimated from a free-decay record, using Equation 19.74. Specifically, the

ratio of the extreme amplitudes in prominent r cycles of decay is determined and substituted into

Equation 19.74 to get the equivalent damping ratio.

Alternatively, if n cycles of damped oscillation are needed for the amplitude to decay by a factor of two,

for example, then, from Equation 19.74, we get

z ¼

2pn

lnð2Þ ¼

0:11

for z , 0:1 ð19:75Þ

For slow decays (low damping), we have

Aiþ1

􀀏 􀀐

ø 2ðAi 2 Aiþ1Þ

ðAi þ Aiþ1Þ ð19:76Þ

Then, from Equation 19.74, we get

z ¼

Ai 2 Aiþ1

pðAi þ Aiþ1Þ

for z , 0:1 ð19:77Þ

Any one of Equation 19.72, Equation 19.74, Equation 19.75, and Equation 19.77 could be employed in

computing z from test data. It should be noted that the results assume single-DoF system behavior.

For multi-DoF systems, the modal damping ratio for each mode can be determined using this method

if the initial excitation is such that the decay takes place primarily in one mode of vibration.

Vibration Damping 19-17

In other words, substantial modal separation and the presence of “real” modes (not “complex” modes

with nonproportional damping) are assumed.

19.4.2 Step – Response Method

This is also a time – response method. If a unit-step

excitation is applied to the single-DoF oscillatory

system given by Equation 19.30, its time – response

is given by

yðtÞ ¼ 1 2

1 ffiffiffiffiffiffiffiffi

1 2 z2

p expð2zvntÞ sin ðvdt þ fÞ

ð19:78Þ

in which f ¼ cos z: A typical step – response curve

is shown in Figure 19.8. The time at the first peak

(peak time), Tp; is given by

Tp ¼

vd ¼

p ffiffiffiffiffiffiffiffi

1 2 z2

vn ð19:79Þ

The response at peak time (peak value), Mp; is

given by

Mp ¼ 1 þ expð2zvnTpÞ ¼ 1 þ exp

2pz ffiffiffiffiffiffiffiffi

1 2 z2

􀁻 !

ð19:80Þ

The percentage overshoot, PO, is given by

PO ¼ ðMp 2 1Þ £ 100% ¼ 100 exp

2pz ffiffiffiffiffiffiffiffi

1 2 z2

􀁻 !

ð19:81Þ

It follows that, if any one parameter of Tp; Mp or PO is known from a step – response record, the

corresponding damping ratio, z; can be computed by using the appropriate relationship from the

following:

z ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2

Tpvn

􀁻 !2

vuut

ð19:82Þ

z ¼

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ

lnðMp 2 1Þ

􀀒 􀀓2

vuuutð19:83Þ

z ¼

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ

lnðPO=100Þ

􀀒 􀀓2

vuuut

ð19:84Þ

It should be noted that when determining Mp the response curve should be normalized to unit steadystate

value. Furthermore, the results are valid only for single-DoF systems and modal excitations in

multi-DoF systems.

19.4.3 Hysteresis Loop Method

For a damped system, the force versus displacement cycle produces a hysteresis loop. Depending on the

inertial and elastic characteristics and other conservative loading conditions (e.g., gravity) in the system,

y (t)

Tp = wd

Unit Step Response

Time t

FIGURE 19.8 A typical step – response of a simple

oscillator.

19-18 Vibration and Shock Handbook

the shape of the hysteresis loop will change. But the work done by conservative forces (e.g., inertial,

elastic, and gravitational) in a complete cycle of motion will be zero. Consequently, the net work done

will be equal to the energy dissipated due to damping only. Accordingly, the area of the displacement –

force hysteresis loop will give the damping capacity, DU (see Equation 19.62). The maximum energy in

the system can also be determined from the displacement – force curve. Then, the loss factor, h; can be

computed using Equation 19.64, and the damping ratio from Equation 19.68. This method of damping

measurement may also be considered basically as a time domain method.

Note that Equation 19.65 is the work done against (i.e., energy dissipation in) a single loading –

unloading cycle per unit mass. It should be recalled that 2zvn ¼ c=m; where c ¼ viscous damping

constant and m ¼ mass. Accordingly, from Equation 19.65, the energy dissipation per unit mass and per

hystereris loop is DU ¼ px2

0vc=m: Hence, without normalizing with respect to mass, the energy

dissipation per hysteresis loop of viscous damping is

DUv ¼ px2

0vc ð19:85Þ

Equation 19.85 can be derived by performing the cyclic integration indicated in Equation 19.62 with the

damping force fd ¼ cx_; harmonic motion x ¼ x0 ejvt and the integration interval t ¼ 0 to 2p=v:

Similarly, in view of Equation 19.51, the energy dissipation per hysteresis loop of hysteretic damping is

DUh ¼ px2

0 h ð19:86Þ

Now, since the initial maximum energy may be represented by the initial maximum potential energy, we

have

Umax ¼ 12

kx2

0 ð19:87Þ

Note that the stiffness, k; may be measured as the average slope of the displacement – force hysteresis loop.

Hence, in view of Equation 19.64, the loss factor for hysteretic damping is given by

h ¼

k ð19:88Þ

Then, from Equation 19.68, the equivalent damping ratio for hysteretic damping is

z ¼

2k ð19:89Þ

Example 19.4

A damping material was tested by applying a loading cycle of 2 900 to 900 N and back to 2 900 N to a

thin bar made of the material and measuring the corresponding deflection. The smoothed load vs.

deflection curve obtained in this experiment is shown in Figure 19.9. Assuming that the damping is

predominantly of the hysteretic type, estimate

1. The hysteretic damping constant

2. The equivalent damping ratio

Solution

Approximating the top and the bottom segments of the hysteresis loop by triangles, we estimate the area

of the loop as

DUh ¼ 2 £ 12

£ 2:5 £ 900 N mm

Alternatively, we may obtain this result by counting the squares within the hysteresis loop. The deflection

amplitude is

x0 ¼ 8:5 mm

Vibration Damping 19-19

Hence, from Equation 19.86 we have

h ¼

2 £

2 £ 2:5 £ 900

p £ 8:52 N=mm ¼ 9:9 N=mm

The stiffness of the damping element is estimated as the average slope of the hysteresis loop; thus

k ¼

600

4:5

N=mm ¼ 133:3 N=mm

Hence, from Equation 19.89, the equivalent damping ratio is

z ¼

9:9

2 £ 133:3

< 0:04

19.4.4 Magnification Factor Method

This is a frequency – response method. Consider a single-DoF oscillatory system with viscous damping.

The magnitude of its frequency – response function is

lHðvÞl ¼

n h

ðv2

n 2 v2Þ2 þ 4z2v2

nv2

i1=2 ð19:90Þ

A plot of this expression with respect to v; the frequency of excitation, is given in Figure 19.10.

The peak value of magnitude occurs when the denominator of the expression is at its minimum.

This corresponds to

ðv2

n 2 v2Þ2 þ 4z2v2

nv2

¼ 0 ð19:91Þ

1000

800

600

400

200

−200

−400

−600

−800

−1000

Force

(N)

Deflection (mm)

−10 −8 −6 −4 −2 0 2 4 6 8 10

FIGURE 19.9 An experimental hysteresis loop of a damping material.

19-20 Vibration and Shock Handbook

The resulting solution for v is termed the resonant frequency, vr

vr ¼

ffiffiffiffiffiffiffiffiffiffi

1 2 2z2

vn ð19:92Þ

It is noted that vr , vd (see Equation 19.70), but for low damping ðz , 0:1Þ; the values of vn; vd; and vr

are nearly equal. The amplification factor, Q; which is the magnitude of the frequency – response function

at resonant frequency, is obtained by substituting Equation 19.92 in Equation 19.90:

Q ¼

ffiffiffiffiffiffiffiffi

1 2 z2

p ð19:93Þ

For low damping ðz , 0:1Þ, we have

Q ¼

2z ð19:94Þ

In fact, Equation 19.94 corresponds to the magnitude of the frequency – response function at v ¼ vn:

It follows that, if the magnitude curve of the frequency – response function (or a Bode plot) is available,

then the system damping ratio, z; can be estimated using Equation 19.94. When using this method, the

frequency – response curve must be normalized so that its magnitude at zero frequency (termed static

gain) is unity.

For a multi-DoF system modal damping values may be estimated from the magnitude of the Bode plot

of its frequency – response function, provided that the modal frequencies are not too closely spaced and

the system is lightly damped. Consider the logarithmic (to the base ten) magnitude plot shown in

Figure 19.11. The magnitude is expressed in decibels (dB), which is calculated by multiplying the

log10(magnitude) by a factor of 20. At the ith resonant frequency, vi; the amplification factor, qi (in dB),

is obtained by drawing an asymptote to the preceding segment of the curve and measuring the peak value

from the asymptote. Then,

Qi ¼ ð10Þqi=20 ð19:95Þ

Magnitude

Frequency ω

ωr =

1−2z 2 wn

H(w)

FIGURE 19.10 The magnification factor method of damping measurement applied to a single-DoF system.

Vibration Damping 19-21

and the modal damping ratio

z ¼

2Qi

; i ¼ 1; 2; …; n ð19:96Þ

If the significant resonances are closely spaced, curve-fitting to a suitable function may be necessary in

order to determine the corresponding modal damping values. The Nyquist plot may also be used in

computing damping using frequency domain data.

19.4.5 Bandwidth Method

The bandwidth method of damping measurement is also based on frequency – response. Consider the

frequency – response function magnitude given by Equation 19.90 for a single-DoF, oscillatory

system with viscous damping. The peak magnitude is given by Equation 19.94 for low damping.

Bandwidth (half-power) is defined as the width of the frequency – response magnitude curve when the

magnitude is ð1=

ffiffi

2 p Þ times the peak value. This is denoted by Dv (see Figure 19.12). An expression

Magnitude (dB)

Asymptotes

w1 w2 w

Frequency

20Log10 H (w)

FIGURE 19.11 Magnification factor method applied to a multi-DoF system.

Magnitude

H(w)

w1 wr w2 Frequency ω

Δw

FIGURE 19.12 Bandwidth method of damping measurement in a single-DoF system.

19-22 Vibration and Shock Handbook

for Dv ¼ v2 2 v1 is obtained below using Equation 19.90. By definition, v1 and v2 are the roots of the

equation

n h

ðv2

n 2 v2Þ2 þ 4z2v2

nv2

i1=2 ¼

1 ffiffi

2 p £ 2z ð19:97Þ

for v: Equation 19.97 can be expressed in the form

v4 2 2ð1 2 2z2Þv2

nv2 þ ð1 2 8z2Þv4

n ¼ 0 ð19:98Þ

This is a quadratic equation in v2; having roots v21

and v22

; which satisfy

ðv2 2 v21

Þðv2 2 v22

Þ ¼ v4 2 ðv21

þ v22

Þv2 þ v21

v22

¼ 0

Consequently,

v21

þ v22

¼ 2ð1 2 2z2Þv2

n ð19:99Þ

and

v21

v22

¼ ð1 2 8z2Þv4

n ð19:100Þ

It follows that

ðv2 2 v1Þ2 ¼ v21

þ v22

2 2v1v2 ¼ 2ð1 2 2z2Þv2

n 2 2

ffiffiffiffiffiffiffiffiffiffi

1 2 8z2

For small z (in comparison to 1), we have

ffiffiffiffiffiffiffiffiffiffi

1 2 8z2

ø 1 2 4z2

Hence,

ðv2 2 v1Þ2 ø 4z2v2

or, for low damping

Dv ¼ 2zvn ¼ 2zvr ð19:101Þ

From Equation 19.101 it follows that the damping ratio can be estimated from the bandwidth using the

relation

z ¼

vr ð19:102Þ

For a multi-DoF system with widely spaced resonances, the foregoing method can be extended

to estimate modal damping. Consider the frequency – response magnitude plot (in dB) shown in

Figure 19.13.

Since a factor of

ffiffi

2 p corresponds to 3 dB, the bandwidth corresponding to a resonance is given by the

width of the magnitude plot at 3 dB below that resonant peak. For the ith mode, the damping ratio is

given by

zi ¼

Dvi

vi ð19:103Þ

The bandwidth method of damping measurement indicates that the bandwidth at a resonance is a

measure of the energy dissipation in the system in the neighborhood of that resonance. The simplified

Vibration Damping 19-23

relationship given by Equation 19.103 is valid for low damping, however, and is based on linear system

analysis. Several methods of damping measurement are summarized in Box 19.2.

19.4.6 General Remarks

There are limitations to the use of damping values that are experimentally determined. For example,

consider time – response methods for determining the modal damping of a device for higher modes. The

customary procedure is to first excite the system at the desired resonant frequency, using a harmonic

exciter, and then to release the excitation mechanism. In the resulting transient vibration, however, there

invariably will be modal interactions, except in the case of proportional damping. In this type of test, it is

tacitly assumed that the device can be excited in the particular mode. In essence, proportional damping is

assumed in modal damping measurements. This introduces a certain amount of error into the measured

damping values.

Expressions used in computing damping parameters from test measurements are usually based on

linear system theory. However, all practical devices exhibit some nonlinear behavior. If the degree of

nonlinearity is high, the measured damping values will not be representative of the actual behavior

of the system. Furthermore, testing to determine damping is usually performed at low amplitudes of

vibration. The corresponding responses could be an order of magnitude lower than, for instance, the

amplitudes exhibited under extreme operating conditions. Damping in practical devices increases

with the amplitude of motion, except for relatively low amplitudes (see Figure 19.14 illustrating

nonlinear behavior). Consequently, the damping values determined from experiments should be

extrapolated when they are used to study the behavior of the system under various operating

conditions. Alternatively, damping could be associated with a stress level in the device. Different

components in a device are subjected to varying levels of stress, however, and it might be difficult to

obtain a representative stress value for the entire device. One of the methods recommended for

estimating damping in structures under seismic disturbances, for example, is by analyzing earthquake

response records for structures that are similar to the one being considered. Some typical damping

ratios that are applicable under operating basis earthquake (OBE) and safe-shutdown earthquake

(SSE) conditions for a range of items are given in Table 19.6.

When damping values are estimated using frequency – response magnitude curves, accuracy becomes

poor at very low damping ratios ð, 1%Þ: The main reason for this is the difficulty in obtaining a

sufficient number of points in the magnitude curve near a poorly damped resonance when the

frequency – response function is determined experimentally. As a result, the magnitude curve is poorly

defined in the neighborhood of a weakly damped resonance. For low damping ð, 2%Þ; time – response

methods are particularly useful. At high damping values, the rate of decay can be so fast that the

measurements contain large errors. Modal interference in closely spaced modes can also affect measured

damping results.

H(w)

3dB

Frequency w

Magnitude (dB)

20Log10

Dwi

FIGURE 19.13 Bandwidth method of damping measurement in a multi-DoF system.

19-24 Vibration and Shock Handbook

Box 19.2

DAMPING MEASUREMENT METHODS

Method Measurements Formulas

Logarithmic decrement

method

Ai ¼ first significant amplitude;

Aiþr ¼ amplitude after r cycles

Logarithmic decrement

d ¼

Aiþr ðper cycleÞ

2p ¼

z ffiffiffiffiffiffiffiffi

1 2 z2

p ðper radianÞ

or,

z ¼

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ ð2p=dÞ2

For low damping

z ¼

Ai 2 Aiþ1

pðAi þ Aiþ1 Þ

Step response

method

Mp ¼ first peak value normalized r.t.

steady-state value;

PO ¼ percentage overshoot

(over steady-state value)

Mp ¼ 1 þ exp

2pz ffiffiffiffiffiffiffiffi

1 2 z2

" #

PO ¼ 100 exp

2pz ffiffiffiffiffiffiffiffi

1 2 z2

" #

Hysteresis loop

method

DU ¼ area of displacement – force

hysteresis loop;

x0 ¼ maximum displacement of the

hysteresis loop;

k ¼ average slope of the

hysteresis loop

Hysteretic damping constant

h ¼

px2

Loss factor

h ¼

Equivalent damping ratio

z ¼

Magnification factor

method

Q ¼ amplification at resonance,

w.r.t. zero-frequency value

Q ¼

ffiffiffiffiffiffiffiffi

1 2 z2

For low damping

z ¼

Bandwidth method Dv ¼ bandwidth at 1=

ffiffi

2 p

of resonant peak

(i.e., half-power bandwidth);

vr ¼ resonant frequency

z ¼

2vr

Vibration Damping 19-25

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