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34.2 Vibration Basics

Back

The vibration phenomena that manifest in rotating machinery can be divided into two major categories:

forced vibration and self-excited instability. A stimulus or a source of excitation is required to initiate and

sustain vibratory motion in a rotor. When the stimulus is a forcing phenomenon such as mass unbalance,

it will produce forced flexural vibration in the rotor analogous to linear forced vibration response in a

simple spring – mass system. On the other hand, self-excited vibration (instability) does not require a

forcing phenomenon for its initiation or sustenance. A description of these phenomena is given next.

34.2.1 Forced Vibration

A rotating force vector (unbalance), a steady

directional force (gravity), or a periodic force

(pump impeller/diffuser interface action), will

cause forced vibration in a rotating machine. The

response of the rotor will depend on the nature of

the forcing function and how it relates to rotor

characteristics. The rotor responses to the most

common excitation phenomena are examined

below.

34.2.1.1 Unbalance Response —

Synchronous Whirling

As an introduction to the theory on rotating

machinery vibration and understanding unbalance

response, it is most appropriate to examine

Jeffcott’s (1919) rotor, which is a simple model

that has many of the basic characteristics of more

complex rotating machinery. The Jeffcott rotor

represents a massless elastic shaft supported freely

in bearings at its ends and carrying a disk of mass

m at the center of its span. The mass center of the

disk is eccentric to its geometric center by a

distance e. Refer to Figure 34.1.

C ¼ geometric center of the disk

b ¼ phase angle

TABLE 34.1 (continued)

Year Contributor Description

1978 – 1980 Benckert, H.

and Wachter, J.

A method to calculate flow induced spring constants for labyrinth gas

seals and the use of swirl breaks to reduce the destabilizing force

caused by tangential velocity in labyrinth seals was introduced

by them

1980 Nelson, H. He further developed a finite element model of a rotor to include shear

deflection and axial torque effects

1980 Brennen, C. et al. They recognized the presence of substantial shroud forces, which

influences the rotor dynamics of a pump

1986 Muszynska, A. She demonstrated that oil whirl occurs at about one-half the running

speed in a vertical rotor. With further increase in speed, oil whip will

commence when the whirl frequency approaches the critical speed of

the rotor

C M

FIGURE 34.1 Jeffcott rotor.

34-6 Vibration and Shock Handbook

M ¼ mass center of the disk

c ¼ viscous damping coefficient of on the rotor

O ¼ bearing center

r ¼ deflection of rotor from origin

u ¼ angle of precession

k ¼ shaft stiffness

v ¼ angular velocity of the rotor ¼ u_ þ b_

Whirling is defined as the angular velocity of rotation of the rotor geometric center ðcÞ or the time

derivative ð u_Þ of the angle of precession ðuÞ (also see Chapter 32). Synchronous whirling is when the rate

of whirling, u_; is equal to the total angular velocity, v; of the system.

Applying Newton’s Laws of motion to the rotor, the differential equations of motion in polar

coordinates ðr; uÞ are obtained as

r€ þ

r þ

2u_ 2

􀀏 􀀐

r ¼ ev2 cos b ð34:1Þ

ru€þ

r þ 2r

􀀏 􀀐

u_ ¼ ev2 sin b ð34:2Þ

For a steady-state condition, the values of r; b; u_; and v are constant. For synchronous whirling,

Equation 34.1 and Equation 34.2 reduce to

2 v2

􀀏 􀀐

r ¼ ev2 cos b ð34:3Þ

vr ¼ ev2 sin b ð34:4Þ

From Equation 34.3 and Equation 34.4

r ¼

ev2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 v2

􀀏 􀀐2

s ð34:5Þ

b ¼ tan21 cv

2 v2

􀀏 􀀐 ð34:6Þ

F ¼

2 ¼

kev2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 v2

􀀏 􀀐2

s ð34:7Þ

Using the following relationships:

vN ¼

ffiffiffiffi

— Natural frequency of rotor without damping

ccr ¼ 2

ffiffiffiffi

km p — Critical damping coefficient

z ¼

ccr

— Damping ratio

Equation 34.6 and Equation 34.7 are reduced to the following nondimensional form:

e ¼

ke ¼ ðv=vNÞ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð1 2 ðv=vNÞ2Þ2 þ ð2zv=vNÞ2

p ð34:8Þ

b ¼ tan21 2zðv=vNÞ

ð1 2 ðv=vNÞ2Þ ð34:9Þ

Vibration in Rotating Machinery 34-7

Figure 34.2 is a graphical representation of the unbalance response of the rotor as a function of rotating

speed, v: Upon examination of the phase relationship, it is important to note that the phase angle, b;

changes from approximately 08 at low speed to values approaching 1808 at the higher speed. At vN;

b ¼ 908: A pictorial illustration of this phenomenon is given in Figure 34.3.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 0.5 1 1.5 2 2.5 3

Speed Ratio w/wN

Amplification Factor r/e

Damping Ratio = 0

0.1

0.15

0.25

0.4 0.5 0.707

1.0

2.0

4.0

FIGURE 34.2 Jeffcott rotor response with mass eccentricity — amplification vs. speed.

0.00

20.00

40.00

60.00

80.00

100.00

120.00

140.00

160.00

180.00

200.00

0 0.5 1 1.5 2 2.5 3

Speed Ratio w /wN

Phase Angle Deg.

Damping Ratio = 0.01

0.15

0.25

0.5

0.707 1.0 2.0 4.0

FIGURE 34.3 Jeffcott rotor response to unbalance — phase angle vs. speed.

34-8 Vibration and Shock Handbook

For the case of zero damping, when v ¼ vN; the rotor deflection and the bearing forces are

unbounded. For all other cases, the rotor deflection and the bearing forces are bounded, and their

amplitude depends on the damping ratio. If a shaft is quickly accelerated through its critical speed to a

higher working speed, then there may not be enough time for large rotor deflection to take place. At high

speeds, v .. vN ; the amplitude of the rotor deflection decreases and approaches the value e; the

eccentricity of the rotor.

The critical speed, vcr; of a rotor in the general case, is the speed at which the rotor deflection

amplitude or the force amplitude transmitted to the bearings is a maximum.

This implies that, at v ¼ vcr

dv ¼

dv ¼ 0

Using Equation 34.8, the following relationship between the natural frequency of the rotor and its

critical speed is derived:

vcr ¼

ffiffivffiffiffiNffiffiffiffiffi

1 2 2z 2

p ð34:10Þ

From Equation 34.10, it is evident that the critical speed of a rotor is not a fixed value and is dependent

on the degree of rotor damping. When z ¼ 1=

ffiffi

2 p ; the system is said to be critically damped.

It is important to note that rotor response to unbalance (or imbalance) is recognizable and

controllable. The amplitude of the force transmitted to the bearing can be reduced by operation at speeds

above the critical speed, reducing unbalance, increasing viscous damping, and avoiding operation close

to critical speeds.

34.2.1.2 Shaft Bow

A rotor with a bent shaft will behave in a similar manner to a rotor with an eccentric mass (Ehrich, 1999).

At high rotor speeds ðv .. vcrÞ; the shaft will tend to correct the bow as illustrated in Figure 34.4. When

shaft bow is combined with mass eccentricity, unique behavior patterns are produced depending on the

phase angle between the bow and the eccentric mass (Childs, 1993).

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 0.5 1 1.5 2 2.5 3 Amplification Factor r/s

Damping Ratio = 0

0.1

0.15

0.25

0.4

0.5

0.707

4.0 2.0 1.0

Speed Ratio w/wN

FIGURE 34.4 Jeffcott rotor response with shaft bow — amplification vs. speed.

Vibration in Rotating Machinery 34-9

34.2.1.3 Gravity Critical

A special case of synchronous whirling may occur in certain types of horizontal rotors due to the

gravitational force. It is a secondary critical speed commonly called the gravity critical, which can occur in

a very heavy lightly damped rotor. The critical speed will occur at approximately half the natural

frequency of the rotor and its amplitudes of deflection at the critical speed are bounded and

approximately twice the static deflection of the rotor (Gunter, 1966).

34.2.1.4 The Influence of Rotor Inertia and Gyroscopic Action

The effect of rotor inertia is ignored in the Jeffcott model. However, in practice, it is recognized that rotor

inertia and gyroscopic action has an influence on the natural frequencies, critical speeds, and unbalance

response of the rotor, including reverse whirling. In the case of the natural frequency of the rotor (zero

speed), the diametral or rotary inertia provides an additional natural frequency associated with the

rotational degree of freedom (DoF). Also, the inertia effect lowers the first natural frequency (Childs,

1993). In the rotating case, the effect of inertia generates both forward and reverse whirling critical speeds

(Childs, 1993). These forward whirling critical speeds tend to be higher (stiffening effect) and the reverse

whirling critical speed lower than the natural frequency of the rotor. At the forward critical speeds, large

amplitude whirling motion due to imbalance occurs, whereas the reverse critical speeds are insensitive to

imbalance of the rotor.

34.2.1.5 Rotor Housing Response across an Annular Clearance

If the rotor deflection due to imbalance exceeds the uniform annular gap, continuous contact would

occur between the rotor and stator resulting in coupled motion between the rotor and stator (Childs,

1993). For low contact frictional forces, synchronous forward whirling driven by the imbalance forces

will occur. If the contact friction force is large enough to prevent slipping between the rotor and stator,

reverse whirling will take place. For the case of synchronous forward whirling in a certain range of

running speeds, instability will occur due to engagement between the rotor and stator (Black, 1968). The

zones of instability depend on the coupled natural frequency of the rotor and stator and the degree of

rotor deflection with respect to the annular gap.

34.2.1.6 Effect of Nonlinearity and Asymmetry on Forced Vibration Response

The foregoing analysis has assumed that stiffness and damping are linear and symmetric and the resulting

forces are proportional to the deflection and velocity of the rotor. However, in reality, rotating machinery

components have inherent nonlinearities and asymmetries that can have a profound influence on their

rotordynamic behavior. At large amplitudes of motion, stiffness and damping coefficients become

nonlinear and result in modifying the response amplitude and critical speeds of the rotor. Nonlinearity in

the support stiffness will introduce considerable distortion to the otherwise simple harmonic vibration

behavior of a purely linear system. The stiffness and damping coefficients of the bearings and their

supports are asymmetric in most cases, in particular in horizontal machines. As a result, the forced

vibratory responses in the two principal directions are different and can behave independent of each

other. Each principal direction will display a critical speed unique to itself. Ehrich (1999) has presented a

discussion on how nonlinearity and asymmetry of stator systems influence forced vibration response.

The influence of rotor stiffness asymmetry and inertia asymmetry on rotor stability is discussed in

Section 34.2.3.

34.2.2 Self-Excited Vibration

Instability (nonsynchronous whirling) is a self-induced excitation phenomenon, sometimes described as

sustained transient motion, that can occur in rotating machinery. At the inception of instability, the rotor

deflection will continue to build up with increase in speed, whereas in the case of a critical speed

resonance, the amplitude of the deflection reaches a maximum value and then decreases. If the rotor

speed is increased above the instability threshold speed, the large amplitudes of motion will normally

34-10 Vibration and Shock Handbook

result in damage to the machine. Unlike forced flexural vibration, rotor instability is self-induced and

does not require a sustained forcing phenomenon to initiate or maintain the motion. It is known to occur

only in machines operating at speeds well above the critical speeds of the rotor. Furthermore, the rotor

whirling speed is different to the rotor speed (nonsynchronous whirling), and it is identical to the critical

speed irrespective of the rotor speed.

In general, the rotor instability is associated with the existence of a tangential force vector, Fu ; acting at

right angles to the deflection vector and directly opposing the damping force vector, as illustrated in

Figure 34.5. The nature of Fu is such that its magnitude increases proportionately with the rotor

deflection. At the point where Fu equals the external damping force, rotor instability will commence due

to the nullification of the stabilizing force. This will produce a whirling motion of ever increasing

amplitude. Several phenomena inherent in the rotor system that generates such tangential force vectors

have been identified and are discussed below. Rotordynamists believe that there still remain more such

phenomena to be discovered.

34.2.2.1 Internal Friction Damping

This type of instability was first experienced in the early 1920s in blast furnace compressors made by the

General Electric Company. These machines were subject to occasional fits of violent vibration. Newkirk

(1924) carried out a series of experiments to understand the unusual behavior of these machines. Based

on the internal friction theory of Kimball (1924). Newkirk (1924) concluded that the interfacial friction

damping forces at the disk shaft interface caused the subsynchronous whirling.

In order to understand the internal friction damping phenomena let us examine the shaft stresses in

the whirling Jeffcott rotor (Figure 34.1). Figure 34.6 is a cross section of the shaft disk interface. Owing

to its deflection, all of the fibers in the right half of the cross section are in tension, Te; and those in the left

half are in compression, Ce: These fiber stresses tend to straighten the shaft and produce a restoring force,

Fr; which opposes the centrifugal force, m_u r: Furthermore, a set of frictional forces are generated at the

shaft disk interface due to stretching and compression of the fibers. The fibers in the bottom lower half

will be stretched and are under frictional tension, Tf ; and those in the upper half are being compressed

under frictional compression Cf : Similarly to the reaction force, Fr; produced from right to left by Te and

Ce; a reaction force, Fu ; from bottom to top will be produced by the frictional stresses Tf and Cf :

The disturbing force, Fu ; is in the same direction as the whirling motion and as a result will increase as

Destabilizing

force Fq

Rotation

Bearing axis

Whirl

direction

mw2r

cwr

2mwr

cr kr

FIGURE 34.5 Rotor instability — general case.

Vibration in Rotating Machinery 34-11

the whirling motion increases. The force, Fu; will oppose the external damping force, cu_r: At the

threshold of instability, the two forces nullify each other. It is also know that the frequency of whirling, u_;

at the threshold of instability equals the natural frequency, vn; of the rotor. Mathematically it can be

expressed as follows:

Fu ¼ cirðv 2u_Þ ð34:11Þ

where ci is the rotor internal damping coefficient:

cu_r ¼ cirðv 2u_Þ ð34:12Þ

u_ ¼ vN ð34:13Þ

Equation 34.12 and Equation 34.13 yield the following relationship between the threshold speed of

instability, the first critical speed, and the damping factors (both internal and external):

vN ¼ 1 þ

ci ð34:14Þ

34.2.2.2 Tip Clearance Excitation (Alford’s Force, Steam Whirl)

Thomas (1958) investigated the instability of steam turbines and suggested that nonsymmetric

radial clearances caused by an eccentric rotor could result in destabilizing forces, and called them

clearance excitation forces. Subsequently Alford (1965) discovered a similar phenomenon in aircraft gas

turbines and, as a result, the destabilizing forces are sometimes referred to as Alford forces in

North America.

The destabilizing force is created as a result of the variation in the gap between the blade tip and the

stator. When the gap decreases, the leakage decreases and consequently the efficiency increases, resulting

in a torque higher than the average torque produced by a uniform gap. When the gap increases, there is a

corresponding decrease in the torque relative to the average. The variation in torque produced by the

eccentricity results in a tangential force, which is normal to the radial deflection and is in the direction of

the whirling motion as shown in Figure 34.7. Furthermore, it has been illustrated that the magnitude

of the resulting force increases proportionately with the increase in rotor deflection, that is, the decrease

Whirl radius

Rotation

Whirl path

Shaft cross section

Te E

FIGURE 34.6 Internal friction damping forces acting on a rotor.

34-12 Vibration and Shock Handbook

in the gap. The resulting force is a destabilizing force that will oppose the external damping force, and at

some point when they balance each other, rotor instability will occur. A detailed analysis of the tip

clearance forces has been made by Urlichs (1977).

34.2.2.3 Impeller Diffuser Excitation Forces

Instability experienced in centrifugal compressors provides a strong suspicion that impeller– diffuser

interaction phenomena are involved in the development of destabilizing forces. However, to date, no

satisfactory destabilizing mechanism or source involving impellers has been identified. In the last two

decades there have been several studies related to rotordynamic forces arising from shrouded centrifugal

pump impellers, but very little work has been done on compressor impellers. The destabilizing force

arising from the impeller– diffuser/volute interaction of a pump has been determined to be relatively

small (Jery et al., 1984; Bolleter et al., 1985; Adkins and Brennen, 1986; Ohashi et al., 1986). The major

portion of the destabilizing force is known to be generated in the narrow gap region between the casing

and shroud of the impeller (Childs, 1986; Bolleter et al., 1989; Baskharone et al., 1994; Moore and

Palazzolo, 2001). In the case of centrifugal compressors, an empirical method to determine stability of

multistage machines has been proposed (Kirk and Donald, 1983). The stability maps proposed by them

for flow through and back-to-back centrifugal compressors are shown in Figure 34.8 and Figure 34.9.

34.2.2.4 Propeller Whirl

Propeller whirl (Taylor and Browne, 1938; Houbolt and Reed, 1961) is another form of instability which

occurs in aircraft rotors when there is a mismatch in the angular velocity vector of the propeller and

the linear velocity vector of the aircraft. This angular mismatch results in the generation of a moment

whose vector has a component of significant magnitude, which contributes to the instability of the

Destabilizing

force Fq

Rotation

Bearing axis

Increase force due to low blade

clearance

Decreased force due to

large blade clearance

cwr

mw 2r

Whirl

direction

FIGURE 34.7 Tip clearance excitation (Alford forces).

Vibration in Rotating Machinery 34-13

propeller (Vance, 1988). Its magnitude is proportional to both the angular mismatch and the linear speed

of the aircraft. With increasing speed, the magnitude of the destabilizing moment will exceed the rotor

viscous damping moment and result in propeller instability (refer to Figure 34.5). Since the propeller is

supported only from one end, the whirling motion is conical and is found to be in the reverse direction to

propeller rotation. The instability is sensitive to the velocity and density of the air and not a function of

the torque of the machine.

34.2.2.5 Fluid Trapped in a Hollow Rotor

Wolf (1968) has demonstrated that trapped fluid inside a hollow rotor can produce a force component

tangential to the whirl orbit due to viscous drag forces. Under subsynchronous whirling speeds, this force

component acts in the same direction of whirling motion and its magnitude is proportional to the rotor

deflection. With reference to Figure 34.5, this force has all the markings of a destabilizing force, which can

produce instability in the rotor. The threshold speed of instability is reached when the whirling speed

equals the first critical speed of the rotor. It has been shown (Ehrich, 1999) that, at the threshold of

instability, the rotor speed is less than twice the first critical speed. This results in a ratio of whirl speed to

rotor speed in the range of 0.5 to 1.0.

34.2.2.6 Dry Friction Rubs

In Section 34.2.1.5, a dry friction rub situation was identified, where slipping was prevented between the

rotor and stator under contact conditions. The contact was made possible by the deflection of the rotor

104

103

102

101

1.0 2.0 3.0 4.0

CRITICAL SPEED RATIO, N/NCR (DIM.)

PRESSURE PARAMETER, P2ΔP/1000 (LB2/IN.4)

EXPERIENCE LIMIT

FOR AUTHORS' COMPANY

(STABLE DESIGNS)

PRESSURE PARAMETER

VERSUS

SPEED RATIO

P2ΔP

1000

NCR

• CENTRIFUGAL COMPRESSORS

• NO PROBLEMS FROM AERODYNAMIC

EXCITED INSTABILITIES

• FOR UNITS WITH N/NCR ≥ 2.0

FIGURE 34.8 Proposed stability map for flow through centrifugal compressors. (Source: Rotor Dynamical

Instability, 1983. With permission.)

34-14 Vibration and Shock Handbook

due to unbalance forces. When contact is made between the rotor and stator, Coulomb friction produces

a tangential force in the direction opposite to shaft rotation. Since the frictional force prevents slipping,

whirling in the reverse direction to rotation occurs. The whirling speed is equal to rv=C; where r is the

radius of the rotor, C the radial gap, and v the speed of the rotor. Since the frictional force is in the same

direction of whirling, it will cause the magnitude of whirling to increase, resulting in further increase of

the frictional force. When the magnitude of this force exceeds the viscous damping force, rotor instability

will occur. Another possibility is for the dry friction whirling speed to approach the coupled natural

frequency of the rotor and stator, in which case unstable motion termed dry friction whipping takes place

(Ehrich, 1999).

In addition to the case described above, dry friction rubs can occur in journal bearings, seals, wear

rings, or any situation where a small clearance between a rotor and stator exists. The inadvertent closure

of the clearance due to unbalance or lack of proper lubrication can initiate dry friction rub induced

instability in these cases as well.

34.2.2.7 Torque Whirl/Load Torque

When the rotor disk axis is not aligned with the bearing axis, as in the case with an overhung rotor, Vance

(1988) has shown that nonsynchronous whirling (torque whirl) can occur as a result of the misalignment

between the load torque and the driving torque. His findings are based on the analysis of a simple

105

EKOFISK FINAL

(STABLE)

EKOFISK ORIGINAL

(UNSTABLE)

UNCCEPTABLE

(BACK TO BACK

DESIGN)

KAYBOB ORIGINAL

(UNSTABLE)

KAYBOB FINAL

(STABLE)

ACCEPTABLE

PRESSURE PARAMETR

VERSUS

SPEED RATIO

104

102

103

P2 = ΔP/1000 VS N/NCR

P2 = ΔDISCHARGE PRESSURE (PSIA)

ΔP = PRESSURE RISE (PSI)

N = COMPRESSOR OPERATING SPEED (RPM)

NCR = COMPRESSOR RIGID BEARING CRITICAL (PRM)

CRITICAL SPEED RATIO. N/NCR (DIM.)

1.0 2.0 3.0 4.0

PRESSURE PARAMETER, P2ΔP/1000 (LB2/ IN.4)

FIGURE 34.9 Proposed stability map for back-to-back centrifugal compressors. (Source: Rotor Dynamical

Instability, 1983. With permission.)

Vibration in Rotating Machinery 34-15

rotor model. It appears that torque whirl instability can occur only in the case of long slender shafts with

high-load torque values. The practical implications of this theory are still to be fully explored.

34.2.2.8 Oil Whirl/Whip

Newkirk and Taylor (1925) first experienced shaft whipping due to oil action in journal bearings during

their investigation into internal friction induced whirling of rotors. They found that under certain

conditions, a rotor mounted on journal bearings whipped when the rotor was running at any speed above

double the critical speed; the whirling motion was in the forward direction and its speed matched the

critical speed of the rotor. He provided a qualitative explanation of the phenomenon based on the fact

that the oil film rotates at half the velocity of the shaft due to friction drag. Hence, for rotational speeds

near twice the critical speed, the oil film provides the stimulus as its speed matches the critical speed value

resulting in large displacements and whipping. Others have also drawn similar conclusions based on the

suggestion of an oil wedge rotating at half speed, or rotating fluid force fields at half shaft speed. However,

the foregoing fails to explain why oil whip persists at speeds greater than twice the critical speed. Ehrich

(1999) has also provided a qualitative explanation for oil whirl based on the general theory of rotor

instability.

Although a comprehensive explanation of the physical phenomena of oil whirl is still outstanding,

numerous analytical models to identify where it could be encountered have been suggested. Gunter

(1966) has analytically demonstrated that the instability in a rotor supported on journal bearings can be

attributed to the cross coupling bearing coefficients. As a result, most of the research on oil whirl

instability has narrowed to accurate estimation of bearing cross coupling coefficients.

34.2.2.9 Influence of Bearings and Supports on Rotor Instability

The results of the Jeffcott model can be easily adopted to include bearing stiffness and bearing support

stiffness effects, provided they are both linear and circumferentially symmetric (isotropic). For this

particular case, the rotor stiffness, k; is the equivalent stiffness resulting from the series connection of the

shaft, bearings, and support stiffness. The resulting values for v N and vcr will be less than those for the

simply supported Jeffcott model. This will result in lowering the threshold speed of instability.

If the bearing stiffness or the bearing-support stiffness is not symmetric (orthotropic) then it can be

shown (Childs, 1993) that the threshold speed of instability is increased and the maximum amplitude of

deflection of the rotor is reduced in comparison to the case with symmetric bearings.

The effect of damping at the bearings or at the bearing-support is very similar to the influence of

stiffness. It reduces the amplitude of the synchronous rotor response at the critical speed, and elevates the

threshold speed of instability. However, there is a limit to the amount of damping that can be applied.

Excessive damping causes a reduction in stability (Childs, 1993).

The mass of the bearings plays a significant role on rotor stability. If the bearing mass is significantly

larger than the rotor mass, the threshold speed of instability is lowered.

34.2.3 Parametric Instability

The instability phenomena described in Section 34.2.2 can be represented by linear differential equations

where the system parameters such as mass, inertia, stiffness, damping, and natural frequency are assumed

to be constants. There is another subcategory of self-excited motion, referred to as parametric instability,

since it is induced by the periodic variation of the system parameters such as inertia, mass, and stiffness. A

discussion of the more common forms of this phenomenon follows.

34.2.3.1 Shaft Stiffness Asymmetry

If the shaft of a rotor contains a sufficient level of stiffness asymmetry in the two principal axis of flexure,

rotor instability could occur. Smith (1933) investigated the rotor behavior under unsymmetrical

flexibility of the bearing supports and unsymmetrical transverse flexibility of the shaft, taking into

consideration the damping effects as well. The following conclusions were derived based on his

investigation:

34-16 Vibration and Shock Handbook

In the presence of stiffness asymmetry, the onset speed of internal-friction induced instability is

lowered.

When there is no external damping, the rotor becomes unstable at all speeds between the two

undamped natural frequencies in the two orthogonal directions. However, if external damping is

significant, parametric instability may be eliminated.

Within the unstable range, the whirling motion is in the forward direction and is synchronous with the

shaft speed. Further, unlike in the case of internal-friction induced instability; it is theoretically possible

to run through parametric instability. This makes parametric instability quite similar to the case of

unbalance response.

When the asymmetric rotor is acted upon by a transverse disturbing steady force such as gravitational

force, the rotor whirls at twice the speed of the shaft. This motion exhibits a resonant increase in

amplitude at a speed that is approximately half the mean of the two natural frequencies.

34.2.3.2 Rotor Inertia Asymmetry

Crandall and Brosens (1961) analyzed the parametric excitation of a rotor with nonsymmetrical principal

moments of inertia. Their results indicate that the rotor behavior is very similar to the case of rotors

with stiffness asymmetry described in Section 34.2.3.1, and parametric instability over similar speed

ranges occurs.

34.2.3.3 Pulsating Torque

Constant torque acting on a rotor is known to lower its critical speeds because they effectively reduce the

rotor’s lateral stiffness. A pulsating torque introduces lateral vibrations and instabilities into a rotor.

When a combination of a pulsating and constant torque is applied to a rotor, it will induce unstable

lateral vibrations in a specific range of rotor speeds and certain combinations of torque amplitudes. In

the region of unstable lateral motion, the whirling speed of the rotor will coincide with the first critical

speed of the rotor regardless of the rotor speed or the frequency of the pulsating torque. At rotor speeds

outside the unstable region, the whirling speed of the rotor will be coincident with the pulsating torque

frequency.

34.2.3.4 Pulsating Longitudinal Loads

Pulsating axial forces on a shaft that are in the order of magnitude of the buckling load will effectively

cause a periodic variation in its lateral stiffness. This will result in a proportionate reduction of the lateral

natural frequency of the shaft. Therefore, pulsating axial loads are capable of inducing parametric

instability in a shaft for both the rotating and the stationary cases.

34.2.3.5 Nonsymmetric Clearance Effects

Bentley (1974) recognized that large subsynchronous whirling can occur in rotating machinery due to

certain types of nonsymmetric clearance conditions. One such condition is when a rotor’s whirling

motion causes rubbing with a stationary surface over a portion of the rotor orbit. This effectively results

in an increase in the rotor stiffness during the contact portion of the orbit, producing a periodic variation

in rotor stiffness during each cycle. Another situation that produces cyclic variation in rotor stiffness can

occur in the case of a rotor supported on antifriction bearings mounted with a clearance fit to the

housing. The cyclic variation of the effective rotor stiffness produces a subsynchronous whirling motion

at exactly half the rotational speed. Occasionally, whirling at one third and one fourth the running speed

has also been observed by Childs (1993). Instability will occur when the whirling speed corresponds to a

critical speed of the rotor.

34.2.4 Torsional Vibration

Rotating machinery with rotors that have relatively large moments of inertia are susceptible to torsional

vibration problems. Torsional vibration is an oscillatory angular motion that is superimposed on the

steady rotational motion of the rotor. In practice it can easily go undetected, as standard vibration

Vibration in Rotating Machinery 34-17

monitoring equipment is not geared to measure torsional vibration. Special equipment must be used to

detect torsional vibration in rotating machinery. Since the introduction of electric motor variable

frequency drives (VFD), the incidence of field torsional vibration problems has increased. This has been

attributed to the inherent torsional excitation forces present in current designs of VFDs. An added

complication when using VFDs, as compared with using fixed speed drives, is the requirement of

eliminating torsional natural frequencies over a wider speed range. Large synchronous electric motors are

known to produce a large pulsating torque at a frequency that changes from twice the line frequency at

the start, down to zero at the synchronous operating speed. In this case, any torsional natural frequency

between zero and twice the line frequency may be subject to excitation. Most industries have come to

recognize that torsional vibration is a potential hazard, and therefore, needs to be investigated at the

design stage of rotating machinery. Several standard design specifications now require that torsional

analysis is part of the design procedure.

The standard design practice in modeling the system for torsional analysis has been to calculate the

undamped eigenvalues of the rotor as a free body in space. This practice is acceptable for most types of

rotating machinery since the torsional stiffness and damping of bearings is insignificant. Also, in most

cases, the torsional damping of the rotors itself is extremely low. Although the absence of damping is

favorable from an analysis point of view, it makes it extremely difficult to eliminate a torsional vibration

problem when encountered. This deficiency has been partly addressed with the introduction of several

new lines of couplings that have a significant degree of torsional damping. Although not commonly used

in rotating machinery, several torsional dampers such as the Lanchester damper have been developed for

use on reciprocating machines. Dampers of similar design could be developed for use in rotating

machinery to solve torsional vibration problems.

The torsional critical speeds of a simple rotor with one or two DoFs can be calculated by analytical

methods. Numerical methods are used to calculate critical speeds and mode shapes of more complex

systems with higher DoF. The Holzer numerical method (Holzer, 1921) described in Section 34.3.1.11

is one of the common methods used for these analyses.

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