70811_39.pdf

CHAPTER 39, PART I

BALANCING OF

ROTATING MACHINERY

Douglas G. Stadelbauer

INTRODUCTION

The demanding requirements placed on modern rotating machines and equip-

ment—for example, electric motors and generators, turbines, compressors, and

blowers—have introduced a trend toward higher speeds and more stringent accept-

able vibration levels. At lower speeds, the design of most rotors presents few prob-

lems which cannot be solved by relatively simple means, even for installations in

vibration-sensitive environments. At higher speeds, which are sometimes in the

range of tens of thousands of revolutions per minute, the design of rotors can be an

engineering challenge which requires sophisticated solutions of interrelated prob-

lems in mechanical design, balancing procedures, bearing design, and the stability

of the complete assembly. This has made balancing a first-order engineering prob-

lem from conceptual design through the final assembly and operation of modern

machines.

This chapter describes some important aspects of balancing, such as the basic

principles of the process by which an optimum state of balance is achieved in a rotor,

balancing methods and machines, and definitions of balancing terms. The discussion

is limited to those principles, methods, and procedures with which an engineer

should be familiar in order to understand what is meant by “balancing.” Finally, a list

of definitions is presented at the end of it.

In addition to unbalance, there are many other possible sources of vibration in

rotating machinery; some of them are related to or aggravated by unbalance, and so,

under appropriate conditions, they may be of paramount importance. However, this

discussion is limited to the means by which the effect of once-per-revolution com-

ponents of vibration (i.e., the effects due to mass unbalance) can be minimized.

BASIC PRINCIPLES OF BALANCING

Descriptions of the behavior of rigid or flexible rotors are given as introductory

material in standard vibration texts, in the references listed at the end of Part I of this

chapter, and in the few books devoted to balancing. A similar description is included

here for the purpose of examining the principles which govern the behavior of rotors

as their speed of rotation is varied.

39.1

39.2

CHAPTER THIRTY-NINE, PART I

PERFECT BALANCE

Consider a rigid body which is rotating at a uniform speed about one of its three

principal inertia axes. Suppose that the forces which cause the rotation and support

the body are neglected; then it will rotate about this axis without wobbling, i.e., the

principal axis (which is fixed in the body) coincides with a line fixed in space (Fig.

39.1). Now construct circular, concentric journals around the axis at the points where

the axis protrudes from the body, i.e., on the stub shafts whose axes coincide with the

principal axis. Since the axis does not wobble, the newly constructed journals also

will not wobble. Next, place the journals in bearings which are circular and concen-

tric to the principal axis (Fig. 39.2). It is assumed that there is no dynamic action of

the elasticity of the rotor and the lubricant in the bearings. A rigid rotor constructed

and supported in this manner will not wobble; the bearings will exert no forces other

than those necessary to support the weight of the rotor. In this assembly, the radial

distance between the center-of-gravity of the rotor and the shaft axis (i.e., a straight

line connecting the journal axes) is zero. The principal axis and the shaft axis coin-

cide. This rotor is said to be perfectly balanced.

PRINCIPAL AXIS

BEARING

PRINCIPAL AXIS

JOURNAL

FIGURE 39.1

Rigid body rotating about prin-

FIGURE 39.2

Balanced rigid rotor.

cipal axis.

RIGID-ROTOR BALANCING—STATIC UNBALANCE

Rigid-rotor balancing is important because it comprises the majority of the balanc-

ing work done in industry. By far the greatest number of rotors manufactured and

installed in equipment can be classified as “rigid” by definition. All balancing

machines are designed to perform rigid-rotor balancing.*

Consider the case in which the shaft axis is not coincident with the principal axis,

as illustrated in Fig. 39.3. In practice, with even the closest manufacturing tolerances,

the journals are never concentric with

the principal axis of the rotor. If concen-

tric rigid bearings are placed around the

journals, thus forcing the rotor to turn

about the connecting line between the

journals, i.e., the shaft axis, a variable

force is sensed at each bearing.

The center-of-gravity is located on

the principal axis, and is not on the axis

of rotation (shaft axis). From this it fol-

lows that there is a net radial force act-

ing on the rotor which is due to centrifugal acceleration. The magnitude of this force

is given by

PRINCIPAL AXIS

c.g.

BEARING

AXIS OF ROTATION (JOURNAL AXIS)

FIGURE 39.3

Unbalanced rigid rotor.

(39.1)

* Field balancing equipment is specifically excluded from this category since it is designed for use with

both rigid and flexible rotors.

39.3

BALANCING OF ROTATING MACHINERY

where m is the mass of the rotor,

is the eccentricity or radial distance of the center-

of-gravity from the axis of rotation, and

is the rotational speed in radians per sec-

ond. Since the rotor is assumed to be rigid and thus not capable of distortion, this

force is balanced by two reaction forces. There is one force at each bearing. Their

algebraic sum is equal in magnitude and opposite in sense. The relative magnitudes

of the two forces depend, in part, upon the axial position of each bearing with

respect to the center-of-gravity of the rotor. In simplified form, this illustrates the

“balancing problem.” One must choose a practical method of constructing a per-

fectly balanced rotor from this unbalanced rotor.

The center-of-gravity may be moved to the shaft axis (or as close to this axis as is

practical) in one of two ways. The journals may be modified so that the shaft axis and

an axis through the center-of-gravity are moved to essential coincidence. From the-

oretical considerations, this is a valid method of minimizing unbalance caused by the

displacement of the center-of-gravity from the shaft axis, but for practical reasons it

is difficult to accomplish. Instead, it is easier to achieve a radial shift of the center-of-

gravity by adding mass to or subtracting it from the mass of the rotor; this change in

mass takes place in the longitudinal plane which includes the shaft axis and the cen-

ter-of-gravity. From Eq. (39.1), it follows that there can be no net radial force acting

on the rotor at any speed of rotation if

m ′ r = m

(39.2)

where m

is the mass added to or subtracted from that of the rotor and r is the radial

distance to m

′

. There may be a couple, but there is no net force. Correspondingly,

there can be no net bearing reaction. Any residual reactions sensed at the bearings

would be due solely to the couple acting on the rotor.

If this rotor-bearing assembly were supported on a scale having a sufficiently

rapid response to sense the change in force at the speed of rotation of the rotor, no

fluctuations in the magnitude of the force would be observed. The scale would reg-

ister only the dead weight of the rotor-bearing assembly.

This process of effecting essential coincidence between the center-of-gravity of the

rotor and the shaft axis is called “ single-plane ( static ) balancing. ” The latter name for

the process is more descriptive of the end result than of the procedure that is followed.

If a rotor which is supported on two bearings has been balanced statically, the

rotor will not rotate under the influence of gravity alone. It can be rotated to any

position and, if left there, will remain in that position. However, if the rotor has not

been balanced statically, then from any position in which the rotor is initially placed,

it will tend to turn to that position in which the center-of-gravity is lowest.

As indicated below, single-plane balancing can be accomplished most simply (but

not necessarily with great accuracy) by supporting the rotor on flat, horizontal ways

and allowing the center-of-gravity to seek its lowest position. It also can be accom-

plished in a centrifugal balancing machine by sensing and correcting for the unbal-

ance force characterized by Eq. (39.1).

′

RIGID-ROTOR BALANCING—DYNAMIC UNBALANCE

When a rotor is balanced statically, the shaft axis and principal inertia axis may not

coincide; single-plane balancing ensures that the axes have only one common point,

namely, the center-of-gravity. Thus, perfect balance is not achieved. To obtain perfect

balance, the principal axis must be rotated about the center-of-gravity in the longi-

tudinal plane characterized by the shaft axis and the principal axis. This rotation can

39.4

CHAPTER THIRTY-NINE, PART I

be accomplished by modifying the journals (but, as before, this is impractical) or by

adding masses to or subtracting them from the mass of the rotor in the longitudinal

plane characterized by the shaft axis and the principal inertia axis. Although adding

or subtracting a single mass may cause rotation of the principal axis relative to the

shaft axis, it also disturbs the static balance already achieved. From this it can be

deduced that a couple must be applied to the rotor in the longitudinal plane. This is

usually accomplished by adding or subtracting two masses of equal magnitude, one

on each side of the principal axis (so as not to disturb the static balance) and one in

each of two radial planes (so as to produce the necessary rotatory effect). Theoreti-

cally, it is not important which two radial planes are selected since the same rotatory

effect can be achieved with appropriate masses, irrespective of the axial location of

the two planes. Practically, the choice of suitable planes may be important. Usually,

it is best to select planes which are separated axially by as great a distance as possi-

ble in order to minimize the magnitude of the masses required.

The above process of bringing the principal inertial axis of the rotor into essential

coincidence with the shaft axis is called “ two-plane ( dynamic ) balancing. ” If a rotor is

balanced in two planes, then, by definition, it is balanced statically; however, the con-

verse is not true.

FLEXIBLE-ROTOR BALANCING 1

If the bearing supports are rigid, then the forces exerted on the bearings are due

entirely to centrifugal forces caused by the unbalance. Dynamic action of the elas-

ticity of the rotor and the lubricant in the bearings has been ignored.

The portion of the overall problem in which the dynamic action and interaction

of rotor elasticity, bearing elasticity, and damping are considered is called flexible

rotor or modal balancing.

Critical Speed. Consider a long, slender rotor, as shown in Fig. 39.4. It represents

the idealized form of a typical flexible rotor, such as a paper machinery roll or tur-

bogenerator rotor. Assume further that all unbalances occurring along the rotor

caused by machining tolerances, inhomogeneities of material, etc. are compensated

by correction weights placed in the end faces of the rotor, and that the balancing is

done at a low speed as if the rotor were a rigid body.

FIGURE 39.4

Idealized flexible rotor.

Assume there is no damping in the rotor or its bearing supports. Consider a thin

slice of this rotor perpendicular to the shaft axis (see Fig. 39.5 A ). This axis intersects

the slice at its geometric center E when the rotor is not rotating, provided that

deflection due to gravity forces is ignored. The center-of-gravity of the slice is dis-

placed by

from E due to an unbalance in the slice (caused by machining tolerances,

inhomogeneity, etc., mentioned above) which was compensated by correction

weights in the rotor’s end planes. If the rotor starts to rotate about the shaft axis with

an angular speed

, then the slice starts to rotate in its own plane at the same speed

about an axis through E. Centrifugal force m

2 is thus experienced by the slice. This

force occurs in a direction perpendicular to the shaft axis and may be accompanied

δω

39.5

BALANCING OF ROTATING MACHINERY

FIGURE 39.5

Rotor behavior below, at, and above first critical speed.

by similarly caused forces at other cross sections along the rotor; such forces are

likely to vary in magnitude and direction. They cause the rotor to bend, which in turn

causes additional centrifugal forces and further bending of the rotor.

At every speed

, equilibrium conditions require that for one slice, the centrifu-

gal and restoring forces be related by

m (δ+ x )ω

= kx

(39.3)

where x is the deflection of the shaft (the radial distance between the geometric cen-

ter and the shaft axis) and k is the shaft stiffness (Fig. 39.5 B ). In Fig. 39.5, the cen-

trifugal and restoring forces are plotted for various speeds (

ω 1 <ω 2 <ω 3 <ω 4 <ω 5 ).

The point of intersection of the lines representing the two forces denotes the equi-

librium condition for the rotor at the given speeds. For this ideal example, as the

speed increases, the point which denotes equilibrium will move outward until, at say

ω 3 , a speed is reached at which there is no resulting force and the lines are parallel.

Since equilibrium is not possible at this speed, it is called the critical speed. The crit-

ical speed

ω n of a rotating system corresponds to a resonant frequency of the system.

At speeds greater than

ω n ), the lines representing the centrifugal and

restoring forces again intersect. As

ω 3 (

)

increases correspondingly until, for speeds which are large, the deflection x

approaches the value of

increases, the slope of the line m

2 ( x

+δ

, i.e., the rotor tends to rotate about its center-of-gravity.

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