Harris C.M., Piersol A.G. Harris Shock and vibration handbook

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Specific Damping Energy. Another useful measure of material damping is the

amount of energy dissipated per unit volume per cycle, known as the specific damp-

ing energy. For a damping material specimen subject to an applied external force

F(t) = F

cos ωt the specific damping energy D is equal to

D =



F dx = 

2π/ω

Fdt (37.21)

For a viscoelastic material obeying Eq. (37.7)

D = F

kη





(37.22)

But F

= kx





, also from Eq. (37.7), so

D =πx

kη (37.23)

The specific damping energy D increases as the square of the amplitude of vibration

for linear viscoelastic materials, so it is clearly desirable to ensure that the damp-

ing material is strained as vigorously as possible in order to maximize D and hence

the damping of the system. This has an important bearing on the choice of location

within a vibrating system for application of a damping treatment. Furthermore, both

k and η must be as large as possible to ensure maximum energy dissipation in the

system, but this can be done only to the extent that further increases of k and η do

not reduce x

. While D is related to k and η for linear viscoelastic materials, this is

not possible for nonlinear materials or for high cyclic strain levels where nonlinear

behavior occurs; the value of D is then, of itself, often used as a measure of overall

damping performance.

COMPARISON OF DAMPING MEASURES

The damping measures described in this section are related to each other as follows

(Table 37.2):

η=2ζ= = = = = (37.24)

These various equations relate η, ∆, and ζ for viscous and viscoelastic damping of

single degree-of-freedom systems. The relationships usually agree well for low

values of η and ζ (η<0.2 or so), but for higher values the comparisons are not so

precise.

It is important, when analyzing tests to determine the effects of damping treat-

ments on dynamic response, to be consistent in the use of these damping measures

and to recognize that they are not completely equivalent, especially over wide fre-

quency ranges or for multimodal response.

Effects of Mass and Stiffness. Changing the mass or stiffness of a single degree-

of-freedom mass-spring system without changing any other parameters leads to a

change of resonance frequency, and when the frequency changes over a wide range,

the differences of viscous and hysteretic damping become more apparent. For vis-

cous damping, the fraction of critical damping ζ=c/2





changes as k or m change,

∆ω



2πU



2∆



APPLIED DAMPING TREATMENTS 37.17

8434_Harris_37_b.qxd 09/20/2001 12:27 PM Page 37.17

whereas for hysteretic damping η does not change, at least within a limited fre-

quency range.Although viscous and hysteretic damping measures are related by the

simple relationship η=2ζ for a single mode at a particular frequency, they do not

remain equivalent as the frequency changes, and significant differences in response

may be observed.

METHODS FOR MEASURING COMPLEX MODULUS PROPERTIES

Vibrating Beam Test Methods. The vibrating beam test methods are frequently

used to measure the extensional or shear complex modulus properties of damping

materials.

1,6

The dynamic response behavior of the beam, first in the undamped

uncoated form and then with an added damping layer or added constrained configu-

ration, is measured for several modes of vibration and over a range of temperatures.

At each temperature, the measured damped resonance frequency f

, the undamped

resonance frequency f

, and the loss factor η

in the nth mode of vibration are meas-

ured and used in an appropriate set of equations to deduce the Young’s modulus E, or

the shear modulus G, and the loss factor η of the damping material at a number of dis-

crete frequencies and temperatures.

Various configurations of cantilever beams are used to measure viscoelastic

material damping properties in tension-compression or shear at low cyclic strain

amplitudes. Figure 37.11 illustrates some of the configurations used. The damping

layers are bonded to the base beams by means of a stiff adhesive such as an epoxy.

This bonding is very important and must be done well using an adhesive which is

stiff in comparison to the damping layer and is very thin. The thickness ratio h

generally lies in the range 0.1 ≤ h

≤ 2.0, and the length l is about 5 to 10 in. (12.7

to 25.4 cm). The base beam material is typically aluminum, steel, or a stiff epoxy or

epoxy matrix composite material having low intrinsic damping. Great care must be

taken to ensure that the temperature range of the tests is not excessive in relation to

the behavior of the base beam, and in particular to allow for the effect of tempera-

ture on the base beam properties such as Young’s modulus, the resonance frequen-

cies, and the modal loss factor in the absence of the damping layer.The vibration test

is conducted allowing the specimen to soak at each selected temperature for several

minutes, often 30 minutes, to be sure of thermal equilibrium; then the beam is

excited by means of a noncontacting transducer or by impact, and the resulting

response in the frequency domain is measured, either through swept sine-wave exci-

tation or FFT analysis of the transient response signal in the time domain. At each

temperature, several resonance frequencies and modal loss factors are measured

over a wide range of frequencies.The test is then repeated after thermal equilibrium

has been reached at the next selected temperature. The data obtained for the first

mode is usually not used because of the low frequency involved and the high ampli-

tudes and high modal damping of the base beam, as well as because of errors in the

analysis when sandwich beams are used. Such vibrating beam tests are widely used

for measuring viscoelastic material damping properties for shear and extensional

deformation.

9,10

Geiger Thick-Plate Test Method. The Geiger thick-plate method is of impor-

tance because it is widely used to describe damping materials in the automotive

industry. It makes use of a large flat plate, suspended freely from four points selected

to be at or near the nodal lines of the first free-free mode, to which is bonded the

damping layer being evaluated. The rate of decay of vibration amplitude (expressed

in decibels per second) is measured and serves as a measure of the effectiveness of

the damping layer. Figure 37.12 illustrates a typical test setup. The system can be

37.18 CHAPTER THIRTY-SEVEN

8434_Harris_37_b.qxd 09/20/2001 12:27 PM Page 37.18

excited by an impulsive force, measured through a force gage, and applied by a

hammer, by an electromagnetic exciter, an electrically actuated impeller, or by sine-

wave or random excitation. The response can be picked up by an electromagnetic

transducer, in which case cross talk with the excitation transducer must be avoided

by adequate separation or by the use of capacitative or electro-optical transducers

or by a miniature accelerometer. The measured output can be displayed in many

ways, including a decaying sinusoidal trace representing response to an impulsive

excitation (a measure of the logarithmic decrement), or a frequency domain display

in the region of the fundamental free-free mode (loss factor measure).The observed

logarithmic decrement or loss factor value is a measure of the damping of the

plate/damping material system and depends on the plate and treatment thicknesses.

The free-layer treatment equations used for the vibrating beam tests may also be

used with the Geiger plate test provided that the same conditions are satisfied. In

particular, the treatment thickness must be sufficient to make the ratio of the

stiffness of the coated plate to that of the uncoated plate greater than about 1.05.

The size of the specimen and the use of only one mode makes this condition

somewhat less restrictive than for the beam tests, for which the specimens are much

smaller.

APPLIED DAMPING TREATMENTS 37.19

FIGURE 37.11 Cantilever beam damping material test configurations. (A)

Nonsymmetric Oberst beam. (B) Symmetric modified Oberst beam. (C)

Symmetric sandwich beam. (D) Symmetric constrained-layer beam.

8434_Harris_37_b.qxd 09/20/2001 12:27 PM Page 37.19

Single Degree-of-Freedom Resonance Tests. Digital test instrumentation and

data analysis techniques make it relatively easy to conduct vibration tests directly on

relatively small samples of damping materials and to readily determine the damping

properties. Typical test configurations are illustrated in Fig. 37.13. For a resonance

type of test, the specimen is driven inertially by a large vibration table (see Chap.

25), usually by swept sine-wave excitation. The input and output accelerations are

usually measured by accelerometers, and the response parameter of interest is the

amplification A = x/x

as a function of frequency, where x is the amplitude of dis-

placement of the mass and x

is that of the shaker table. At resonance, the maximum

value of A = x/x

is observed along with the resonance frequency ω

for each tem-

perature. The loss factor and modulus in both the tension-compression and shear

loading of the specimen material are determined from

η= (37.25)

E = (37.26)

G = (37.27)

For tension-compression loading, l is the length and S

= wh is the cross-sectional

area of the load-carrying member, where w is the cross-section width and h is the

cross-section thickness, as illustrated in Fig. 37.13. For shear loading, h is the thick-

ness of the shear layer and S

= 2wl is the cross-sectional area of the shear member,

where l is now the breadth of the load-carrying area, again as illustrated in Fig. 37.13.

The effective mass m

includes the added mass m and the effective mass of the spec-

imen damping material, which is about one-third of its actual mass. For the exten-

sional specimen, the ratio l/h or l/w, whichever is smaller, must be greater than 1.0 or

shape effects will have to be taken into account. For the shear configuration, the

ratio h/l must be less than 0.2 for the same reason. For highly damped materials, for

which x/x

does not exceed 1.0 by a significant amount, considerable error in meas-

uring A and η will be encountered, but the method is very effective for values of η

less than about 0.5. In this method, data are obtained at only one frequency; the mass

m must be changed to obtain data at other frequencies. Care must be taken to avoid

sagging or creep of the specimen at high temperatures and to ensure that thermal

equilibrium has been achieved. A thermocouple placed within the volume of the



A

− 1



37.20 CHAPTER THIRTY-SEVEN

FIGURE 37.12 Geiger plate test configuration.

8434_Harris_37_b.qxd 09/20/2001 12:27 PM Page 37.20

specimen material may be necessary, particularly for tests at high strain amplitudes

where internal heating of the specimen by energy dissipation from damping may

lead to wide differences between true specimen temperature and the temperature of

the surroundings.

Impedance Tests. If the specimens are excited by a driver through a force gage,

then the response measure used to characterize the system behavior is the compli-

ance or receptance x/F, where F is the driving force measured by the force gage and

x is the response at the same point, measured by an accelerometer, for example. If

the mass m is large compared with the mass of the specimen, as illustrated in Fig.

37.14, then one may add one-third the mass of the specimen to m to give the effec-

tive mass m

of the equivalent single degree-of-freedom system, so that

= (37.28)

If this is expressed instead in terms of the ratio F/x, the dynamic stiffness at the driv-

ing point, which is directly related to the driving-point impedance, then

κ=k − m

+ jkη (37.29)

which shows that the direct dynamic stiffness is a linear function of ω

and the quad-

rature dynamic stiffness κ

= kη. It is not difficult to obtain good measurements of k



k(1 + jη) − m



APPLIED DAMPING TREATMENTS 37.21

FIGURE 37.14 Impedance test concepts.

FIGURE 37.13 Resonance test concepts.

8434_Harris_37_b.qxd 09/20/2001 12:27 PM Page 37.21

and η by this type of test approach from about 0.2ω

to 3ω

, so data can be obtained

quite easily over about a decade of frequency instead of at only a single frequency as

for the resonance method. Analytical mass corrections may also have to be made to

account for inertial effects at the force gage.

COMMERCIAL TEST SYSTEMS

Many commercial systems are available for measuring the complex modulus prop-

erties of viscoelastic damping materials.

11–13

All are based on some kind of deforma-

tion mode of a sample of the material, measurement of the corresponding excitation

forces and displacements, and analysis of the data to obtain the material properties.

Each system has advantages and disadvantages, but when due care is exercised, good

results usually can be obtained with each system. Particular care should be taken to

read, understand, and follow the manufacturer’s instructions. For example, in some

tests such as monitoring cure cycles of epoxies, the temperature sweep rate can be

quite high in order to keep up with the reaction.This is acceptable if one is monitor-

ing the progress of the cure cycle, but it may not be acceptable if one seeks to mea-

sure the damping properties at a state approximating thermal equilibrium. For

thermal equilibrium to be maintained, temperature sweep rates well below 1°F

(0.5°C) per minute are usually recommended, and even lower rates may be required

for large specimens. A dwell period at each temperature is recommended before

performing the test.

REFERENCES

1. Nashif, A. D., D. I. G. Jones, and J. P. Henderson: “Vibration Damping,” John Wiley & Sons,

Inc., New York, 1985.

2. Nashif, A. D., and T. M. Lewis: Sound & Vibration, 25(7):14 (1991).

3. Williams, M. L.: J. AIAA, 2(5):785 (1964).

4. Bagley, R. L., and P. Torvik: J. Rheology, 27(3):201 (1983).

5. Ferry, J. D.: “Viscoelastic Properties of Polymers,” 3d ed., John Wiley & Sons, Inc., New

York, 1981.

6. Jones, D. I. G.: “Handbook of Viscoelastic Vibration Damping,” John Wiley & Sons, Ltd.,

Chichester, U.K., 2001.

7. Harris, C. M.: “Shock and Vibration Handbook,” 4th ed., The McGraw-Hill Companies,

Inc., New York, 1996.

8. Oberst, H.: Akustica, 4:181 (1952).

9. “DampFEA Beam Test System,” Roush Anatrol Corp., Sunnyvale, Calif.

10. “DTI VBT Beam Test System,” Damping Technologies, Inc., Mishawaka, Ind.

11. “Viscoelasticmeter Impedance Test System,” Metravib S. A., Ecully, France.

12. “DuPont DMP Flexural Impedance Test System,” DuPont Instruments,Inc.,Wilmington,

Del.

13. “PL-DMTA Impedance Test System,” Polymer Laboratories, Ltd., Loughborough, U.K.

37.22 CHAPTER THIRTY-SEVEN

8434_Harris_37_b.qxd 09/20/2001 12:27 PM Page 37.22

CHAPTER 38

TORSIONAL VIBRATION IN

RECIPROCATING AND

ROTATING MACHINES

Ronald L. Eshleman

INTRODUCTION

Torsional vibration is an oscillatory angular motion causing twisting in the shaft of a

system; the oscillatory motion is superimposed on the steady rotational motion of a

rotating/reciprocating machine. Even though the vibration cannot be detected with-

out special measuring equipment, its amplitude can be destructive. For example,

gear sets that alter speeds of power transmission systems transmit the vibration to

the casing. Similarly, slider crank mechanisms in engines and compressors convert

torques to radial forces that are discernable to human perception but are not meas-

urable because of the insensitivity of test equipment and background noise. If gear-

boxes or reciprocating machines are part of a drive train, excess noise and vibration

can indicate trouble. Standards and measurement methods dealing with acceptable

magnitudes of radial vibration are provided in Chap. 19.

Motion is rarely a concern with torsional vibration unless it affects the function of

a system. It is stresses that affect the structural integrity and life of components and

thus determine the allowable magnitude of the torsional vibration.Torsional vibratory

motions can produce stress reversals that cause metal fatigue. Components tolerate

less reversed stress than steady stress. In addition, stress concentration factors associ-

ated with machine members decrease the effectiveness of load-bearing materials.

Figure 38.1 illustrates the twisting of a shaft of an electric motor-compressor sys-

tem. The torsional mode shape associated with the first torsional natural frequency

is shown in Fig. 38.2. A coupling in the power train allows for misalignment in the

assembly. The mode shape shows that the stiffness of the coupling is much less than

that of other shaft sections. This is indicated by the large slope (change in angular

displacement) of the mode shape at the coupling.The coupling will be the predomi-

nant component in the motor-compressor system governing the torsional natural

frequency associated with the mode.

Torsional vibration is usually a complex vibration having many different fre-

quency components. For example, shock resulting from abrupt start-ups and unload-

38.1

8434_Harris_38_b.qxd 09/20/2001 12:25 PM Page 38.1

ing of gear teeth causes transient torsional vibration in some systems; start-up of

synchronous electric motor systems may cause torsional resonance. Random tor-

sional vibrations caused by gear inaccuracies and ball bearing defects are relatively

common in rotating machines.

MODELING

The torsional elastic system of a drive unit and its associated machinery is a com-

plicated arrangement of mass and elastic distribution. The complete mechanical

system can include the drive unit, couplings, gearboxes or other speed-changing

devices, and one or more driven units. This complicated system is made amenable

to mathematical treatment by representing it as a model—a simpler system that

is substantially equivalent dynamically. The equivalent system usually consists of

lumped masses which are connected by massless torsionally elastic springs as illus-

trated in Fig. 38.3. The masses are placed at each crank center and at the center

planes of actual flywheels, rotors, propellers, cranks, gears, impellers, and arma-

tures.

The torsional calculation is made not for the drive unit alone but for the com-

plete system, including all driven machinery. On an engine, it is usually possible to

consider such parts as camshafts, pumps, and blowers either as detached from the

engine (if they are driven elastically) or as additional rigid masses at the point of

attachment to the crankshaft (if the driver is relatively rigid). If there is doubt, these

parts should be included in the torsional calculation as elastically connected masses

and removed if the natural frequencies do not change after the parts are removed

from the model.

38.2 CHAPTER THIRTY-EIGHT

FIGURE 38.1 Schematic drawing illustrating the twisting of

the shaft of a motor-compressor system.

FIGURE 38.2 Torsional mode shape for the motor-

compressor system shown in Fig. 38.1.

8434_Harris_38_b.qxd 09/20/2001 12:26 PM Page 38.2

CALCULATION OF POLAR MOMENTS OF INERTIA

Circular Disc or Cylinder Rotating about a Perpendicular Axis. The polar

moment of inertia, essential in modeling torsional vibration, often is easy to calcu-

late.The general form is J =∫r

dm, where r is the instantaneous radius, and dm is the

differential mass. The formula for the polar moment of inertia of a circular disc or

cylinder rotating about a perpendicular axis is

J = lb-in.-sec

(38.1)

where J = polar moment of inertia,

lb-in.-sec

γ= material density, lb/in.

d = diameter of disc or cylin-

der, in.

l = axial length of disc or

cylinder, in.

g = acceleration due to gravity,

386.1 in./sec

Piston and Connecting Rod. The pis-

ton and connecting rod shown schemati-

cally in Fig. 38.4 introduce a

variable-mass problem, the solution of

which is complex. The exact solution

shows that the effect of the piston and

connecting rod can be closely approxi-

mated by representing them as a concen-

trated rotor of polar inertia J defined by

J =



+ W



1 −



lb-in.-sec

(38.2)

where W

= weight of piston, piston pin, and cooling fluid, lb

= weight of connecting rod, lb

h = fraction of rod length from crank pin to center-of-gravity

R = crank radius, in.



πd

lγ



32g

TORSIONAL VIBRATION IN RECIPROCATING AND ROTATING MACHINES 38.3

FIGURE 38.3 A model of the motor-compressor

system shown in Fig. 38.1, consisting of a series of

masses connected by massless torsionally elastic

springs (K = stiffness, lb⋅in./rad; J = polar moment of

inertia, lb⋅in.⋅sec

FIGURE 38.4 Schematic diagram of a crank

and connecting rod.

8434_Harris_38_b.qxd 09/20/2001 12:26 PM Page 38.3

Crankshaft. The polar inertias of the crank webs (see Fig. 38.5), the crankpin,

and the journal sections are added to that given by Eq. (38.2). These polar inertias

should be calculated with the best

obtainable accuracy. The following pro-

cedure is recommended:

Let the crank web be intercepted by

a series of concentric cylinders of radius

R. The polar inertia of the crank web is

defined by

J =





S dR



+ J

(38.3)

where γ=specific weight of the

crank web, lb/in.

= maximum radius of the

crank web, in.

= radius of base cylinder

(see Fig. 38.5), in.

= polar inertia of portion

of crank web within base

cylinder, lb-in.-sec

The integral in the above expression for

J is the area of the R

S curve between

the values of radii R

and R

For the crank web shown in Fig. 38.5 the area S is defined as S = bRθ/57.3, where

 is measured in degrees. The polar inertia can then be expressed as

J =





 dR



+ J

lb-in.-sec

(38.4)

The same procedure is used to calculate the polar inertia of propellers and other

irregular parts. In a marine propeller of ogival sections, i.e., flat driving face, circular

arc back, and elliptically developed outline (do not use for other shapes), the polar

inertia (excluding hub) is given by

J = 0.0046 lb-in.-sec

where n = number of blades

D = diameter of propeller, in.

b = maximum blade width, in.

t = maximum blade thickness at one-half radius (axis to tip), in.

Propellers. For propellers, pumps, and hydraulic couplings an addition must be

made for the virtual inertia of the entrained fluid. For marine propellers this is ordi-

narily assumed at 26 percent of the propeller inertia. Virtual inertias for pumps are

not known accurately, but it can be assumed that half the casing is filled with rotat-

ing fluid.



57.3g



38.4 CHAPTER THIRTY-EIGHT

FIGURE 38.5 View of a crank web in a plane

normal to the crankshaft axis.

8434_Harris_38_b.qxd 09/20/2001 12:26 PM Page 38.4