§6 － 1 Purposes and Methods of Balancing

§6 － 2 Balancing of Rigid Rotors

Chapter 6 Balancing of MachineryChapter 6 Balancing of Machinery

The center of mass of some machine elements may not coincide with their r

otating centers because of the asymmetry of the structure. Even for symmetrical

machine elements, the center of mass may still be eccentric because of uneven dist

ribution of materials, errors in machining and also in casting and forging. The cen

trifugal force exerted on the frame is time varying, and it will therefore impart vib

ration to the frame. This vibration can adversely affect the structural integrity of t

he machine foundation.

一、 Purposes of Balancing

Purposes ： We should try to eliminate the unwanted centrifugal forces in

machines, especially in high-speed machinery and precision machinery.

二、 Methods

1. Balancing of rotors

Rotors ——Parts constrained to rotate about a fixed axis are called rotors.

§§66 －－ 1 1 Purposes and Methods of Balancing

（ 1 ） Rigid rotors

Rigid rotors——If the rotating frequency of the rotor is less than (0.

6 - 0 .7) nCl (where: nCl is the first resonant frequency of the rotor), t

hen the rotor is supposed to have no deformation during rotation a

nd is called a rigid rotor.

Flexible rotors——If the working rotating frequency of the rotor is

larger than (0.6-0.7) nCl, then the rotor will have large elastic defor

mation due to imbalance during rotation. The elastic deformation

makes the eccentricity larger than the original one so that a new im

balance factor is added and the balancing problem becomes more c

omplicated. Such a rotor is called a flexible rotor.

（ 2 ） Flexible rotors

2. Balancing of mechanisms

The coupler of a linkage has a complex motion. The accelerati

on of its mass center and its angular acceleration vary throughout t

he motion cycle. The coupler will therefore create a varying inertia

force and inertia moment of force for any mass distribution. So the

balance of link ages must be considered as a whole. The resultant in

ertia force of all moving parts is equal to the net unbalanced force a

cting on the frame of a machine, which is referred to as the shaking

force. Likewise, a resultant unbalanced moment acting on the fram

e, caused by the inertia forces and inertia moments of all moving pa

rts, is called the shaking moment. The shaking force and the shakin

g moment will cause the frame to vibrate.

Rotors whose axial dimensions B are small compared to their diameters D

(usually B /D < 0.2), the masses of such rotors are assumed practically to lie in a

common transverse plane.

一、 Calculation for the Balancing of a Rigid Rotor

ω

D

B

e

GG

FI

All centrifugal forces in this disk-like rotor are planar and concurrent. If the v

ector sum of these forces is zero, then the mass center of the system coincides with

the shaft center and the rotor is balanced. Otherwise, it is called imbalance. Since t

he imbalance can be shown statically, such imbalance is called static imbalance.

§§66 －－ 22 Balancing of Rigid Rotors

O x

y

r1 m1

1

r2

m2

2

rb

mb

b

Unbalanced masses are depicted as point masses mi at radial di

stances ri. In this case, there are two masses, m1 and m2. When the rotor rotates with constant angular v

elocity ω, each of the unbalanced masses produces a c

entrifugal force FIi

FIi = miω2ri

In this case, a third mass mb with rotating

radius of rb is added to the system so that the

vector sum of the three centrifugal forces is

zero and balance is achieved.

m1r1 + m2r2+ mbrb = 0 ∑ F = ∑FIi ＋ Fb= 0

(mbrb )x = - ∑ miricosi (mbrb )y = - ∑ mirisini

mbrb= (mbrb )x2

+(mbrb )y2 b= acrtan[(mbrb )y / (mbrb )x]

r’b m’

b

If b / D >0.2 ， although the resultant of

the two centrifugal forces is zero, the forces are

not collinear and a resultant couple will exist.

The direction of the resultant couple changes

during rotation. The resultant couple will act on the frame and tend to produce

rotational vibration of the frame. Such an imbalance can only be detected by me

ans of a dynamic test in which the rotor is spinning. Therefore, this is referred t

o as dynamic imbalance. The rotor in Fig. is therefore statically balanced and d

ynamically unbalanced.

二、 Dynamic Balancing of Rigid Rotors

From the above, we can see that the conditions for the balancing

of a non-disk rigid rotor are: Both the vector sum of all inertia

forces and the vector sum of all moments of inertia forces about

any point must be zero.

Calculation for the Dynamic Balancing

L

I

II

F2IF1I

F3I

F2II

F3II

F1II

m2

m3m1

r1

F2

r2

F3

r3

F1

l1

l2

l3

From Theoretical Mechanics, we know that the centrifugal force F can be repl

aced dynamically by a pair of forces F Ⅰ and FⅡ parallel to F and acting in two ar

bitrarily chosen transverse planes and . In this way, the complicated spatial Ⅰ Ⅱforce system has been converted into two simpler planar concurrent force systems

on two planes.

FL

lFI

1

FL

lLFII

1

The above methods can be extended to any rotor w

ith any number of imbalances on any number of transv

erse planes. The conclusion is that any number of masse

s on any number of transverse planes of a non-disk rigi

d rotor can be balanced dynamically by a minimum of t

wo masses placed in any two arbitrarily selected transve

rse planes. The selected transverse planes are called bal

ance planes. In practice, those planes on which counter

weights can be mounted easily ,or mass can be removed

easily, may be chosen as the balance planes.