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STATIC AND DYNAM IC BALANCING

OBJECTIVE

The objectives of this experiment are:

(1) To demonstrate experimentally the differences between static and dynamic balancing.

(2) To verify the vectorial method of calculation of the positions of counterbalancing

weights in rotating mass systems.

DISCUSSION

Inertia forces exist wherever parts having mass are accelerated. The forces are important internally

because the parts themselves must be designed to perform satisfactorily under all combinations of inertiaand service loads. They are also important externally because the resulting external or shaking force

becomes a disturbing force on the supporting frame and associated parts. In both cases varying forces

acting on elastic bodies can give rise to serious, even destructive, vibrations of the parts or complete

machine and adjacent structures and equipment. The presence of vibration and the accompanying noise can

be serious problems with respect to the physical and mental well-being of operators and others.

The general approach to the minimization of the magnitude of the inertial shaking forces is to

balance the effect by introducing another shaking force that, in so far as possible, is equal in magnitude and

opposite in direction to the original shaking force. This process is called balancing.

THEORY

r

aelement of mass

dM

mass, M

rotation axis

axialreferencepoint

Figure 1. Rotating Mass

As the terms imply, static unbalance refers to an object at rest and dynamic unbalance refers to a

rotating body. A body is statically balanced if it has no tendency to rotate about its axis of rotation when at

rest. This condition must be satisfied no matter what orientation the body is put in. The condition for

static balance is simply that the axis of rotation passes through the center of gravity of the body. Thus,

static balancing requires only that:


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Static and Dynamic Balancing 2

r dMM

=z 0 (1)

where r is the perpendicular distance from the rotation axis to the elemental mass, dM. The integral is

integrated over the entire mass, M. Static balance can always be achieved by making only one correctionand the amount of correction is independent of the plane in which it is to be made.

A mass that is not dynamically balanced will not be apparent until it is rotated -- then the axis of

rotation will have a tendency to wobble about the mass center. Dynamic balance requires not only that the

axis of rotation passes through the center of gravity [equation (1)] but also that it be a principal axis of

inertia. This second requirement is satisfied if:

r a dMM

=z 0 (2)

where a is the axial distance along the rotation axis from some arbitrary location. Corrections in twoseparate planes are required to dynamically balance a mass.

A rotating system can often be separated into a number of discrete masses, each with their own

unbalances. Each of the masses can then be represented by concentrated mass located at its own center ofmass. An example of this is shown in Figure 2.

axis of rotation

mr1

mr2

mr3

mr4

mr3

mr2

mr4

mr1

End ViewIsometric View

Figure 2. Discrete Mass System

a1

a4

a3

a2

arbitrary reference point

A discrete mass system can be statically balanced by satisfying the vector equation:

m ri i

i

n

=

=

v

1

0 (3)

where n is the total number of discrete masses. For a static balance only one correction mass is required.To dynamically balance the system two equations must be satisfied:

m ri ii

n

=

=

v

1

0 and m r ai i ii

n

=

=

v

1

0 (4)

Two balancing masses must be used to dynamically balance a discrete mass system.


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PROCEDURE

Determi nation of Mass Moments of Bl ocks

(1) Level the apparatus using the four leveling feet and the bubble level. With the main frame of the

apparatus rigidly fixed to the top of the support frame at right angles, the motor drive belt is

removed to allow the shaft to rotate freely. Remove the blocks from the shaft by removing thesocket head cap screws and sliding them to one end of the shaft.

(2) Use the cord and container system to determine the mass moment of each block in terms of the

"weight of the steel balls." This is the weight of the steel balls necessary to rotate the blocks from

a vertical to a horizontal position.

(a) Wrap the cord one and a half times around the disk at the end of the shaft so that it will not

slip.

(b) Place block #1 on the shaft and secure it using the screws.

(c) Add steel balls to one of the containers, one at a time, while lightly tapping the frame (to

overcome bearing friction). Add balls until the block is horizontal.

(d) Weigh the balls on a scale to determine a magnitude of unbalance.

(e) Repeat steps (b) through (d) for the remaining three blocks.

mi ri (weight of steel bal ls)

Block #1 #2 #3 #4

Unbalance

Static Balance(3) The rotating system is to be statically balanced. Blocks #1 and #2 will be used to represent an

unbalanced shaft and Blocks #3 and #4 will be used as correction masses to first statically balance

the system and then to dynamically balance the system.

(a) Position Block #1 at 1.5" and 0o and Block #2 at 4.5" and 60o to represent the unbalanced

system. The axial position is measured using the scale on the side of the frame and the

vernier scale.

(b) Using the vector polygon method, determine the angular orientations of Blocks #3 and #4

required for static balance. Equation (3) indicates that the following equation must be

satisfied:m r m r m r m r

1 1 2 2 3 3 4 40 + + + =

v v v v

The magnitude of each vector was determined in the previous step. The angles of the first

two vectors was also specified. To achieve a condition of static balance the angles of the

last two blocks must be determined. This is done graphically as shown in Figure 3. Use

a protractor to measure the determined angles, 3 and 4.

(c) Place blocks #3 and #4 in the center of the shaft at the determined angles, 3 and 4respectively. Check the accuracy of the static balance (i.e., see if the shaft has any

tendency to rotate by itself). Mount the shaft in running position (suspended from

springs) and attach drive belt. Run the shaft and observe operation.


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m r1 1

m r2 2

80

m r3 3

m r4 4

magnitude of

magnitude of

Determinedsolution

4

3

Figure 3. Graphical Solution for Static Balance

(1) Draw known vector #1.

(2) Add vector #2 to vector #1.

(head to tail)

(3) Make an arc centered at tip

of #2 with the radius of #3.

(4) Make an arc centered at tail

of #1 with the radius of #4.

(5) Arc intersection specifies the

required directions of #3 and #4.

Intersection

Dynamic Balance

(4) The rotating system is to be dynamically balanced The same angular positions of Blocks #3 and#4 as that determined in the static balancing procedure will be used (so that static balancing will be

maintained). It is now necessary to compute the axial positions of Blocks #3 and #4 so that

dynamic balancing is achieved.

(a) Check that the known blocks are placed in the correct positions: Block #1 at 1.5" and 0o

and Block #2 at 4.5" and 60o. The axial position is measured using the scale on the side

of the frame and the vernier scale.

(b) Determine the known information about the vectors to be used in the second equation in

equation (4). Summarize the information in the table below.

Block #1 #2 #3 #4

miri [from table in step 2(e)]

ai [given or to be determined] 1.50" 4.50" ???? ????

miriai [computed] ???? ????

i[given or from static balance]

(c) Using the vector polygon method, determine the axial positions of Blocks #3 and #4

required for dynamic balance. Equation (4) indicates that the following equation must be

satisfied:m r a m r a m r a m r a1 1 1 2 2 2 3 3 3 4 4 4 0 + + + =

v v v v

The angles of the first two vectors was specified. The angles of the second two vectors

was determined in achieving a static balance. To dynamically balance the system, the

axial distance locations (a3 and a4) of the last two blocks must be determined. This is

done graphically as shown in Figure 4. Use a scale to measure the m3r3a3 and m4r4a4vectors. Determine ai by dividing miriaiby the appropriate miri value.


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m r a1 1 1

m r a2 2 2

80

m r a3 3 3

m r a4 4 4

direction of

direction of

Determinedsolution

Figure 4. Graphical Solution for Dynamic Balance

(1) Draw known vector #1.

(2) Add vector #2 to vector #1.

(head to tail)(3) Draw the direction of #3

through the tip of #2.

(4) Draw the direction of #4

through the tail of #1.

(5) The line intersection of vectors

#3 and #4 determines their length.

Intersection

(d) Place blocks #3 and #4 at the determined axial position on the shaft, a3 and a4

respectively. Check the accuracy of the dynamic balance by running the shaft andobserving its operation.

REPORT

The report will be a "short report" consisting of the following:

1. Title page (title of experiment, name, test date, course).

2. Abstract (about 100 words summarizing what was done and the most important conclusions).

3. Results - what values were determined as being necessary for static and dynamic balancing.

4. Conclusions - brief, specific, factual, including any disagreement with theory, reasons for

disagreement including sources of experimental errors).

5. Computations - including vector polygons (carefully drawn), and equations and computations

necessary to determine dynamic balance.

6. Discussion of applying this particular method of dynamic balancing to engineering parts.

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