32.6 Whirling of Shafts

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In the previous two sections, we studied the vibration excitations caused on rotating shafts and their

bearings due to some form of mass eccentricity. Methods of balancing these systems to eliminate the

undesirable effects were also presented. One limitation of the given analysis is the assumptions that the

rotating shaft is rigid and, thus, does not deflect from its axis of rotation due to the unbalanced

excitations. In practice, however, rotating shafts are made lighter than the components they carry (rotors,

disks, gears, etc.) and will undergo some deflection due to the unbalanced loading. As a result, the shaft

will bow out and this will further increase the mass eccentricity and associated unbalanced excitations

and gyroscopic forces of the rotating elements (disks, rotors, etc.). The nature of damping of rotating

machinery (which is rather complex and incorporates effects of rotation at bearings, structural

deflections, and lateral speeds) will further affect the dynamic behavior of the shaft under these

conditions. In this context, the topic of whirling of rotating shafts becomes relevant.

Consider a shaft that is driven at a constant angular speed v (e.g., by using a motor or some other

actuator). The central axis of the shaft (passing through its bearings) will bow out. The deflected axis

itself will rotate, and this rotation is termed whirling or whipping. The whirling speed is not necessarily

equal to the drive speed v (at which the shaft rotates about its axis with respect to a fixed frame).

However, when the whirling speed is equal to v; the condition is called synchronous whirl, and the

associated deflection of the shaft can be quite excessive and damaging.

To develop an analytical basis for whirling, consider a light shaft supported on two bearings carrying a

disk of mass m in between the bearings, as shown in Figure 32.19(a). Note that C is the point on the disk

at which it is mounted on the shaft. Originally, in the neutral configuration when the shaft is not driven

ðv ¼ 0Þ; the point C coincides with point O on the axis joining the two bearings. If the shaft were rigid,

then the points C and O would continue to coincide during motion. The mass center (centroid or center

of gravity for constant g) of the disk is denoted as G in Figure 32.19. During motion, C will move away

from O due to the shaft deflection. The whirling speed (speed of rotation of the shaft axis) is the speed of

rotation of the radial line OC with respect to a fixed reference. Denoting the angle of OC with respect to a

fixed reference as u; the whirling speed is u_: This is explained in Figure 32.19(b) where an end view of the

Vibration Design and Control 32-33

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disk is given under deflected conditions. The constant drive speed v of the shaft is the speed of the shaft

spin with respect to a fixed reference, and is the speed of rotation of the radial line CG with respect to the

fixed horizontal line shown in Figure 32.19(b). Hence, the angle of shaft spin is vt; as measured with

respect to this line. The angle of whirl, u; is also measured from the direction of this fixed line, as shown.

32.6.1 Equations of Motion

Under practical conditions, the disk moves entirely in a single plane. Hence, its complete set of equations

of motion consists of two equations for translatory (planar) motion of the centroid (with lumped mass

m), and one equation for rotational motion about the fixed bearing axis. The latter equation depends on

the motor torque that drives the shaft at constant speed v; and is not of interest in the present context.

Thus, we will limit our development to the two translatory equations of motion. The equations may be

written either in a Cartesian coordinate system ðx; yÞ or a polar coordinate system ðr; uÞ: Here, we will use

the polar coordinate system.

Consider a coordinate frame (i, j) that is fixed to the disk with its i axis lying along OC as shown

in Figure 32.19(b). Note that the angular speed of this frame is u_ (about the k axis that is orthogonal to i

and j). Hence, as is well known, we have

di

dt ¼ u_j and

dj

dt ¼ 2_ ui ð32:69Þ

The position vector of the mass point G from O is

OGk ¼ rG ¼ OCk þ CGk ¼ ri þ e cosðvt 2uÞi þ e sinðvt 2uÞj ð32:70Þ

The velocity vector vG of the mass point G can be obtained simply by differentiating Equation 32.70 with

the use of Equation 32.69. However, this can be simplified because v is constant. Here, line CG has a

velocity ev that is perpendicular to it about C. This can be resolved along the axes i and j. Hence, the

velocity of G relative to C is

vG=C ¼ 2ev sinðvt 2 uÞi þ ev cosðvt 2 uÞj

However, the velocity of point C is

vC ¼

d

dt

ri ¼ ri þ r

di

dt ¼ ri þ ru_j

Hence, the velocity of G, which is given by vG ¼ vC þ vG=C; can be expressed as

vG ¼ r i þ ru_ j 2 ev sinðvt 2uÞi þ ev cosðvt 2uÞj ð32:71Þ

(a)

r θ

O

Disk with

Mass Eccentricity

OC = r

CG = e

ω t

(b)

O

C

m

G

Shaft

Bearing

b, bf

k

C

G

j

i

e

FIGURE 32.19 (a) A whirling shaft carrying a disk with mass eccentricity; (b) end view of the disk and whirling

shaft.

32-34 Vibration and Shock Handbook

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Similarly, the acceleration of C is

aC ¼

d

dt

vC ¼

d

dt ½ri þ ru_j􀀉 ¼ r€i þ ru_j þ ru_j þ ru_j 2 ru_2i ¼ ðr€ 2 ru_2Þi þ ðru€þ 2ru_Þj

Also, since line CG rotates at constant angular speed v about C, the point G has only a radial (centrifugal)

acceleration ev2 along GC. This can be resolved along i and j as before. Hence, the acceleration of G

relative to C is

aG=C ¼ 2ev2 cosðvt 2 uÞi 2 ev2 sinðvt 2 uÞj

It follows that the acceleration of point G, given by aG ¼ aC þ aG=C; may be expressed as

aG ¼ ðr€ 2 ru_2Þi þ ðru€þ 2ru_Þj 2 ev2 cosðvt 2 uÞi 2 ev2 sinðvt 2 uÞj ð32:72Þ

The forces acting on the disk are as follows:

Restraining elastic force due to lateral deflection of the shaft ¼ 2kri

Viscous damping force (proportional to the velocity of C) ¼ 2bri 2 br_u j

In addition, there is a frictional resistance at the bearing, which is proportional to the reaction and,

hence, the shaft deflection is r and also depends on the spin speed v: The following approximate model

may be used:

Bearing friction force ¼ 2bf rvj

Here,

k ¼ lateral deflection stiffness of the shaft at the location of the disk

b ¼ viscous damping constant for lateral motion of the shaft

bf ¼ bearing frictional coefficient

The overall force acting on the disk is

f ¼ 2b r i 2 ðbr_u þ bf rvÞj ð32:73Þ

The equation of rectilinear motion

f ¼ maG ð32:74Þ

on using Equation 32.72 and Equation 32.73, reduces to the following pair in the i and j directions:

2kr 2 br ¼ m½r€ 2 ru_2 2 ev2 cosðvt 2 uÞ􀀉 ð32:75Þ

2bru_ 2 bf rv ¼ m½ru€þ 2ru_ 2 ev2 sinðvt 2 uÞ􀀉 ð32:76Þ

These equations may be expressed as

r€ þ 2zvvnr þ ðv2

n 2u_2Þr ¼ ev2 cosðvt 2 uÞ ð32:77Þ

ru€þ 2ðzvvnr þ rÞ u_ þ 2zf vnvr ¼ ev2 sinðvt 2 uÞ ð32:78Þ

where the undamped natural frequency of lateral vibration is

vn ¼

ffiffiffiffi

k

m

s

ð32:79Þ

and

zv ¼ viscous damping ratio of lateral motion

zf ¼ frictional damping ratio of the bearings

Equation 32.77 and Equation 32.78, which govern the whirling motion of the shaft-disk system, are a

pair of coupled nonlinear equations, with excitations (depending on v) that are coupled with a motion

variable (u). Hence, a general solution would be rather complex. A relatively simple solution is possible,

however, under steady-state whirling.

Vibration Design and Control 32-35

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32.6.2 Steady-State Whirling

Under steady-state conditions, the whirling speed u_ is constant at u_ ¼ vw; hence, u€ ¼ 0: Also, the lateral

deflection of the shaft is constant, hence, r ¼ r€ ¼ 0: Therefore, Equation 32.77 and Equation 32.78

become

ðv2

n 2 v2

w Þr ¼ ev2 cosðvt 2 uÞ ð32:80Þ

2zvvnvw r þ 2zf vnvr ¼ ev2 sinðvt 2 uÞ ð32:81Þ

In Equation 32.80 and Equation 32.81, the left-hand side is independent of t: Hence, the right-hand side

should also be independent of t: For this, we must have

u ¼ vt 2 f ð32:82Þ

where f is interpreted as the phase lag of whirl with respect to the shaft spin (v), and should be clear

from Figure 32.19(b). It follows from Equation 32.82 that, for steady-state whirl, the whirling speed

u_ ¼ vw is

vw ¼ v ð32:83Þ

This condition is called synchronous whirl because the whirl speed ðvw Þ is equal to the shaft spin speed

ðvÞ: It follows that under steady-state conditions, we should have the state of synchronous whirl. The

equations governing steady-state whirl are

ðv2

n 2 v2Þr ¼ ev2 cos f ð32:84Þ

2zvnvr ¼ ev2 sin f ð32:85Þ

along with Equation 32.82 and, hence, Equation 32.83. Here, z ¼ zv þ zf is the overall damping ratio of

the system. Note that the phase angle f and the shaft deflection r are determined from Equation 32.84

and Equation 32.85. In particular, squaring these two equations and adding to eliminate f; we obtain

r ¼

ev2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðv2

n 2 v2Þ2 þ ð2zvnvÞ2

p ð32:86Þ

which is of the form of magnitude of the frequency transfer function of a simple oscillator with an

acceleration excitation. Divide Equation 32.85 by Equation 32.84 to get the phase angle:

f ¼ tan21 2zvnv

ðv2

n 2 v2Þ ð32:87Þ

Using simple calculus (differentiate the square and equate to zero), we can show that the maximum

deflection occurs at the critical spin speed vc given by

vc ¼

ffiffivffiffinffiffiffiffiffiffi

1 2 2z2

p ð32:88Þ

This critical speed corresponds to a resonance. For light damping, we have approximately vc ¼ vn:

Hence, critical speed for low damping is equal to the undamped natural frequency of bending vibration

of the shaft-rotor unit. The corresponding shaft deflection is (see Equation 32.86)

rc ¼

e

2z ð32:89Þ

which is also a good approximation of r at critical speed, with light damping. From Equation 32.84 and

Equation 32.85, we can see that, at critical speed (with low damping), sin f ¼ 1 and cos f ¼ 0; which

gives f ¼ p=2: Also, note from Equation 32.86 that the steady-state shaft deflection is almost zero at low

speeds and approaches e at very high speeds. However, Equation 32.87 shows that, for small v; tan f

is positive and small. We can see from Equation 32.85 that sin f is positive. This means f itself is small

for small v: For large v; we can see from Equation 32.86 that r approaches e: Thus, we can see from

32-36 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Equation 32.87 that tan f is small and negative, whereas Equation 32.85 shows that sin f is positive.

Hence, f approaches p for large v:

It is seen from Equation 32.89 that, at critical speed, the shaft deflection increases with mass

eccentricity and decreases with damping. This indicates that the approaches for reducing the damaging

effects of whirling are:

1. Eliminate or reduce the mass eccentricity through proper construction practices and balancing.

2. Increase damping.

3. Increase shaft stiffness.

4. Avoid operation near critical speed.

There will be limitations to the use of these approaches, particularly making the shaft stiffer. Note also

that our analysis did not include the mass distribution of the shaft. A Bernoulli – Euler type beam analysis

has to be incorporated for a more accurate analysis of whirling for shafts whose mass cannot be accurately

represented by a single parameter that is lumped at the location of the rotor. Formulas related to whirling

of shafts are summarized in Box 32.5.

Example 32.6

The fan of a ventilation system has a normal operating speed of 3600 rpm. The blade set of the fan weighs

20 kg and is mounted in the mid-span of a relatively light shaft that is supported on lubricated bearings at

its two ends. The bending stiffness of the shaft at the location of the fan is 4.0 £ 106 N/m. Equivalent

damping ratio that acts on the possible whirling motion of the shaft is 0.05. Owing to fabrication error,

the centroid of the fan has an eccentricity of 1.0 cm from the neutral axis of rotation of the shaft:

1. Determine the critical speed of the fan system and the corresponding shaft deflection at the

location of the fan at steady state.

2. What is the steady-state shaft deflection at the fan during normal operation?

The fan was subsequently balanced using a mass of 5 kg. The centroid eccentricity was reduced to

2 mm by this means. What is the shaft deflection at the fan during normal operation now? Comment on

the improvement that has been realized.

Solution

1. The system is lightly damped. Hence, the critical speed is given by the undamped natural frequency;

thus

vc ø vn ¼

ffiffiffiffi

k

m

s

¼

ffiffiffiffiffiffiffiffiffiffiffi

4 £ 106

20

s

rad=sec ¼ 447:2 rad=sec

The corresponding shaft deflection is

rc ¼

e

2z ¼

1:0

2 £ 0:05

cm ¼ 10:0 cm

2. Operating speed v ¼ ð3600=60Þ £ 2p rad=sec ¼ 377 rad=sec: Using Equation 32.86, the corresponding

shaft deflection, at steady state, is

r ¼

1:0 £ ð377Þ2

½ð447:22 2 3772Þ2 þ ð2 £ 0:05 £ 447:2 £ 377Þ2􀀉1=2 cm ¼ 2:36 cm

After balancing, the new eccentricity e ¼ 0:2 cm:

The new natural frequency (undamped) is

vn ¼

ffiffiffiffiffiffiffiffiffiffiffi

4 £ 106

25

s

rad=sec ¼ 400 rad=sec

Vibration Design and Control 32-37

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The corresponding shaft deflection during steady-state operation is

r ¼

0:2 £ ð377Þ2

½ð4002 2 3772Þ2 þ ð2 £ 0:05 £ 400 £ 377Þ2􀀉1=2 cm ¼ 1:216 cm

Note that, even though the eccentricity has been reduced by a factor of five by balancing, the operating

deflection of the shaft has been reduced only by a factor of less than two. The main reason for this is that

the operating speed is close to the critical speed. Methods of improving the performance include

changing the operating speed, using a smaller mass to balance the fan, using more damping, and making

Box 32.5

WHIRLING OF SHAFTS

Whirling: A shaft spinning at speed v about its axis, may bend due to flexure. The bent (bowed

out) axis will rotate at speed vw : This is called whirling.

Equations of motion:

r€ þ 2zvvnr þ ðv2

n 2u_2Þr ¼ ev2 cosðvt 2 uÞ

ru€þ 2ðzvvnr þ rÞ u_ þ 2zf vnvr ¼ ev2 sinðvt 2 uÞ

where ðr; uÞ are polar coordinates of shaft deflection at the mounting point of lumped mass.

e ¼ eccentricity of the lumped mass from the spin axis of shaft

u_ ¼ vw ¼ whirling speed

v ¼ spin speed of shaft

vn ¼

ffiffiffiffiffi

k=m p ¼ natural frequency of bending vibration of shaft

k ¼ bending stiffness of shaft at lumped mass

m ¼ lumped mass

zv ¼ damping ratio of bending motion of shaft

zf ¼ damping ratio of shaft bearings

Steady-state whirling (synchronous whirl):

Here, whirling speed ð u_ or vw Þ is constant and equals the shaft spin speed v (i.e., vw ¼ v for

steady-state whirling).

Shaft deflection at lumped mass

r ¼

ev2

ðv2

n 2 v2Þ2 þ ð2zvnvÞ2

􀀑 􀀜1=2

Phase angle between shaft deflection (r) and mass eccentricity (e)

f ¼ tan21 2zvnv

ðv2

n 2 v2Þ

where z ¼ zv þ zf

Note: For small spin speeds v; we have small r and f: For large v; we have r ø e and f ø p

Critical speed:

Spin speed v ¼

ffiffivffiffinffiffiffiffiffiffi

1 2 2z2

p vn for small z

f ¼ p=2

32-38 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

the shaft stiffer. However, some of these methods may not be feasible. Operating speed is determined by

the task requirements. A location may not be available that is sufficiently distant to place a balancing mass

that is appropriately small. Increased damping will increase heat generating, cause bearing problems, and

will also reduce the operating speed. Replacement or stiffening of the shaft may require too much

modification to the system and add cost. A preferable alternative would be to balance the fan by removing

some mass. This will move the critical frequency (natural frequency) away from the operating speed

rather than closer to it, while reducing the mass eccentricity at the same time. For example, suppose that a

mass of 3 kg is removed from the fan, which results in an eccentricity of 2.0 mm. The new natural/critical

frequency is

ffiffiffiffiffiffiffiffiffiffiffi

4 £ 106

17

s

rad=sec ¼ 485:1 rad=sec

The corresponding shaft deflection during steady operation is

r ¼

0:2 £ ð377Þ2

½ð485:12 2 3772Þ2 þ ð2 £ 0:05 £ 485:1 £ 377Þ2􀀉1=2 cm ¼ 0:3 cm

In this case, the deflection has been reduced by a factor of eight.

32.6.3 Self-Excited Vibrations

Equation 32.77 and Equation 32.78, which represent the general whirling motion of a shaft, are nonlinear

and coupled. In these equations, the motion variables (r and u) occur as (nonlinear) products of the

excitation (v). Such systems are termed self-excited. Note that, in general (before reaching the steady

state) the response variables r and u will exhibit vibratory characteristics in view of the presence of the

excitation functions cosðvt 2 uÞ and sinðvt 2 uÞ: Hence, a whirling shaft may exhibit self-excited

vibrations. Because the excitation forces directly depend on the motion itself, it is possible that a

continuous energy flow into the system could occur. This will result in a steady growth of the motion

amplitudes and represents an unstable behavior.

A simple example of self-excited vibration is provided by a pendulum whose length is time variable.

Although the system is stable when the length is fixed, it can become unstable under conditions of

variable length. Practical examples of self-excited vibrations with possible exhibition of instability

include the flutter of aircraft wings due to coupled aerodynamic forces, wind-induced vibrations of

bridges and tall structures, galloping of ice-covered transmission lines due to air flow-induced vibrations,

and chattering of machine tools due to friction-related excitation forces. Proper design and control

methods, as discussed in this chapter, are important in suppressing self-excited vibrations.

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