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Mili J. Hota & D.P. Vakharia

242

number of excellent texts, but it can be difficult to quickly pull out the practical insight

needed. At the other end of the spectrum, there is also a large number of

troubleshooting resources that focus on identification of problems and characteristics,

but only offer limited insight. Discussion of recent combined experimental and

analytical effort raised the possibility of an article that would attempt to provide a

deeper insight into some of the basic characteristics of rotating machinery vibration

from a less mathematical perspective.

Thus in this article several issues that are basic to an understanding of rotating

machinery vibration are been discussed:

• What are “critical speeds”?

• How do critical speeds relate to resonances and natural frequencies?

• How do natural frequencies change as the shaft rotational speed changes?

•

How are shaft rotational natural frequencies different from more familiarnatural frequencies and modes in structures?

• What effects do bearing characteristics have?

2. Vibration Intuition2.1 A Brief Review of Structural Vibration

As engineers, we learn that vibration characteristics are determined by a structure’s

mass and stiffness values, with damping (ability to dissipate vibrational energy)

playing an integral role by controlling amplitudes. This education generally starts with

the simplest possible system – a rigid mass attached to a spring as shown in Fig. 1[1].

With this simple system, we quantify our intuition about vibrational frequency (heavierobjects result in lower frequency, stiffer springs yield higher frequency). After some

work, we reach the conclusion that the free vibration frequency is controlled by the

square root of the ratio of stiffness to mass as shown in Fig. 1. We could then add a

viscous damper parallel with the spring, and provide a sinusoidal force as shown in

Fig. 2[3]. By carefully applying a constant amplitude sinusoidal force that slowly

increases in frequency and recording the amplitude of the motion, we could then

generate the classic normalized frequency responses of a spring-mass-damper system.

By repeating the test with a variety of dampers, the classic frequency response shown

in Fig. 3 can be developed. Assuming we knew the mass, stiffness and damping of our

system, this response is also predicted quite well by the standard frequency domain

solution to the differential equation of motion for this system shown in Equation 1[2].

222

0

1 ⎟ ⎠

⎞⎜⎝

⎛ +⎟⎟

⎠

⎞⎜⎜⎝

⎛ −

=

k

c

k

m

k

F

Amplitude

ω ω

(1)

There are several noteworthy points about these frequency responses. The first is

that at low excitation frequencies, the response amplitude is roughly constant and

greater than zero. The amplitude is governed by the ratio of the applied force to the



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damp

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and Mode

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ore techni

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frequency

to this fr

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teristics wi

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correspon

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f spring m

plitude fo

cross secti

243

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D.P. Vakh

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ical Speed

Fig. 8

drical Me to do use

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: Effect of

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. Again,

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.” Fig. 7[8]

forward a

the surface

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of the whi

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anges the s

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frequenci

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of Vibrati

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odes.

t’s consid

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ume our s

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mode sha

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motion ha

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mes called

mprope’ m

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shows rot

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tor moves

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ituation, w

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245

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first

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246

frequencies over a wide shaft speed range. This plot is often referred to as a “Campbell

Diagram.” From this figure, we can see that the frequencies of this cylindrical mode do

not change very much over the speed range. The backward whirl mode drops slightly,

and the forward whirl mode increases slightly (most noticeably in the high stiffness

case). The reason for this change will be explored in the next section.

4. Rotating Dynamics – Conical Mode Now that we have explored the cylindrical mode, let’s look at the second set of modes.

Fig. 9 shows the next frequencies and mode-shapes for the three machines. The

frequencies are close to the nonrotating modes where the disk was rocking without

translating. The modes look a lot like the nonrotating modes, but again involve circular

motion rather than planar motion. To visualize how the rotor is moving, imagine

holding a rod stationary in the center, and moving it so that the ends trace out twocircles. The rod traces the outline of two bulging cones pointed at the center of the rod.

Thus, this mode is sometimes referred to as a ‘conical’ mode. Viewed from the side,

the rod appears to be rocking up and down around the center, with the left side being

out-of-phase from that on the right. Thus, this mode is also sometimes called a ‘rock’

mode or a ‘pitch’ mode.

As with the first mode and the nonrotating modes, the low bearing stiffness mode

is generally referred to as a rigid mode, and a high bearing stiffness pulls in the rotor

ends. As with the cylindrical mode, the whirl can be in the same direction as the rotor’s

spin (“forward whirl”), or the opposite direction (“backward whirl”).

To see the effects of changing shaft speeds, we could again perform the

analysis/modal test from non spinning to a high spin speed and follow the two

frequencies associated with the conical mode. Fig. 10 plots the forward (red line) and

backward (black dashed line) natural frequencies over a wide speed range. From this

figure, we can see that the frequencies of the conical modes do change over the speed

range. The backward mode drops in frequency, while the forward mode increases.

The explanation for this surprising behavior is a gyroscopic effect that occurs

whenever the mode shape has an angular (conical/rocking) component. First consider

forward whirl. As shaft speed increases, the gyroscopic effects essentially act like an

increasingly stiff spring on the central disk for the rocking motion. Increasing stiffness

acts to increase the natural frequency. For backward whirl, the effect is reversed.

Increasing rotor spin speed acts to reduce the effective stiffness, thus reducing thenatural frequency (as a side note, the gyroscopic terms are generally written as a skew-

symmetric matrix added to the damping matrix – the net result, though, is a stiffening/

softening effect). In the case of the cylindrical modes, very little effect of the

gyroscopic terms was noted, since the center disk was whirling without any conical

motion. Without the conical motion, the gyroscopic effects do not appear. Thus, for the

soft bearing case, which has a very cylindrical motion, no effect was observed, while

for the stiff bearing case, which has a bulging cylinder (and thus conical type motion

near the bearings), a slight effect was noted.



A R

Fi

2.3

No

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thre

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that we

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a fictitiou

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[11].

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inal model

• The in

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• The inc

point o

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nd freque

mparison o

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).

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roperties, o

ts act to c

conical co

be a nomin

eavy disk

ttached (i.e

longer dis

s smaller

moment Ip

e disk dia

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uencies ve

e that:

de frequen

de unchan

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tural frequ

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247

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248

•

The reduced mass moment of inertia version does not change the first mode

(disk center of gravity has very little conical motion).

•

The reduced mass moment of inertia increases the frequency of the second

mode, and decreases the strength of the gyroscopic effect (disk center of

gravity has substantial conical motion).

For the second case, let’s move the disk to the end, and move the bearing inboard

to result in an overhung rotor with the same mass and overall length. Fig. 14 shows the

three models and the three sets of natural frequencies versus speed. Comparing the

nominal model to the two modified versions, the important things to note are:

The increased mass lowers the first mode frequencies and very slightly lowers the

second mode frequencies.

The reduced mass moment of inertia version increases the frequency of both the

first and second modes, and decreases the strength of the gyroscopic effect.If we looked at the mode shapes and these plots, we would again see that the

reasons are the same as for the center disk rotor. Changes in affect the natural

frequency of that mode but have little effect if it is at a node. Changes to mass moment

of inertia at a location of large whirl orbit, on the other hand, have little effect.

Changes to mass moment of inertia at a node with large conical motions have a strong

effect on the corresponding mode. Although not entirely obvious from the plots

presented, changes in the ratio of polar mass moment of inertia to diametric mass

moment of inertia change the strength of the gyroscopic effect. Indeed, for a very thin

disk (a large ratio), the forward conical mode increases in speed so rapidly that the

frequency will always be greater than the running speed. Indeed, there will be no

conical critical speed as defined below.

2.4 Critical Speeds

With some insight into rotating machinery modes, we can move on to “critical speeds.”

The American Petroleum Institute(API), in API publication 684 (First Edition, 1996),

defines critical speeds and resonances as follows:

• Critical Speed – A shaft rotational speed that corresponds to the peak of a

noncritically damped (amplification factor > 2.5) rotor system resonance

frequency. The frequency location of the critical speed is defined as the

frequency of the peak vibration response as defined by a Bodé plot (for

unbalance excitation).

•

Resonance – The manner in which a rotor vibrates when the frequency of a

harmonic (periodic) forcing function coincides with a natural frequency of the

rotor system.

Thus, whenever the rotor speed passes through a speed where a rotor with the

appropriate unbalance distribution excites a corresponding damped natural frequency,

and the output of a properly placed sensor displays a distinct peak in response versus

speed, the machine has passed through a critical speed. Critical speeds could also be

referred to as “peak response” speeds.



A R

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frequenci

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We can se

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. 3 reveals

sponse equ

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ibution tha

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balance E

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are the re

versus unb

tation was

citation hacitation, w

ence – th

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rs by referr

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from criti

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alance for

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set of unb

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al speed –

ing to Fig.

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first three

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4: Phase r

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, whereas t

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requency

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be a const

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14. This Fi

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. At the cri

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Figures 1

g-mass-da

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249

API

del,

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orce

to a

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disk

, the

sing




250

through the critical speed, the phase between the unbalance force and the response

direction has changed by 180°. As a result, the disk now rotates around the mass center

of the disk/unbalance. Once the disk achieves this state, further increases in speed do

not change the amplitude until the effects of the next mode are observed.

5. ConclusionsIt was shown that cylindrical rotor modes are not influenced by gyroscopic effects and

remain at a fairly constant frequency versus rotor speed. Conversely, conical rotor

modes are indeed influenced and caused to split into forward and backward whirl

components that respectively increase and decrease in frequency with increased rotor

speed.

References

[1]

Michael I. Friswell et al. Dynamics of rotating machines, 1 edition 2010. {79}

[2]

AKPOBI, J.A.; OVUWORIE, “G.C. Computer–Aided Design of the Critical Speed

of Shafts” J. Appl. Sci. Environ. Manage. December, 2008 Vol. 12(4) 79 – 86

[3] Den Hartog, J. P. Mechanical Vibrations, 4th ed. New York:McGraw-Hill,1996.

[4] Eshleman, R. Torsional vibration of machine systems. Proceedings of the 6th Turbo

machinery Symposium, Texas A&M University, 2010, p.13.

[5] Farouk Hamdoon O, “Application of finite element package for modeling rotating

machinery vibration”, Engg and Tech Journal, vol. 27, No 12, 2009.

[6]

Henrich Sprysl, Gunter Ebi, “Bearing stiffness determination through vibration

analysis of shaft line of hydro power plant”, International Journal of hydro power &dams, 1998, pp. 437-447

[7]

John M Vance, Brain T Murphy, “Critical speeds of turbo machinery computer

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