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All due care has been taken to ensure that the contents of thismanual are accurate and up to date. However, if any errors arediscovered please inform TQ so the problem may be rectified.

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TQ Education and Training LtdProducts DivisionPE/MC/O295

KEEP FOR FUTURE REFERENCE

PRODUCT: TMI02 STATIC AND DYNAMIC BALANCING APPARATUS

In compliance with the EC directive on Safety of Machinery, the followinginformation should be noted:

This equipment is only to be used in accordance with instructions in themanual. Students using the equipment must be adequately supervised. Localregulations regarding the use of electricity, gasoline, diesel oil, kerosene,mercury must be observed in using this apparatus.

Foreseen use of machineDemonstration / experimentation on static and dynamic balancing.

Installation and assembly instructionsThe equipment is supplied fully assembled for bench mounting. It is designedfor use with a O-l2V DC electrical supply, e.g. the Te<.'Quipment E66 SpeedController. H using another electrical supply it is recommended that themaximum current supplied is limited to lOA.

Operating instructionsConnect the apparatus to a suitable power supply using the terminals next tothe on-off switch. A microswitch is fitted to the apparatus so that it will onlyfunction when the perspex cover is in position. The on-off switch must be heldin the 'on' position for the motor to function.Note: The motor and rotating shaft assembly is mounted on a metal platewhich may be clamped down. Ensure the plate is clamped fully down forstatic balancing tests. For dynamic balancing release the clamps, turn through90° and lock down, leaving the plate free to vibrate.The equipment should be used within its operating limits (see operatingconditions ).

Maintenance and inspectionNo user maintenance necessary other than to periodically inspect electricalleads and connections for wear. Periodically check that all warning labels arein position and legible.

Handling instructionsNet weight: 17 kg.Ensure the correct procedures for handling the above weight are used whenmoving this apparatus.

Operating Conditions

Noise LevelThe measured sound pressure level of this apparatus is less than 70 dB(A).

(i)

1...I5.T. Of CQNT.EN.TS-.

Section

LIST OF SYMBOLS(11)

11. INTROOUCTION

THEORY2. 3

2.12.22.32.4

3

6

9

12

Stat;c BalanceDynam;c BalanceDynam;c Balanc;ng of 3 MassesDynam;c Balanc;ng of More Than 3 Masses

3.- EXPERIMENTAL APPARATUS 5

EXPERIMENTAL PROCEDURE AND TYPICAL RESULTS4. 7

4.1 7

174.24.34.4

Demonstrat;on of Stat;c Balance but DynamicInbalance

S;mple Dynam;c Balance us;ng 4 ~'assesStat;c and Dynam;c Balance of a 4 Mass SystemTyp;cal Calculat;ons

19

2~

(1;)

LIST OF SYMBOLS

shaft-wise location of unbalanced mass or block fromconvenient datum position

a

radius of unbalanced mass from centreline of shaft..

shaft-wise location of blocks relative to zero onscale

x

(W = mg)w weight of unbalanced mass

F centrifugal force acting on unbalanced massW(F = m~2 or - nA>2)g

angular displacement of mass from convenient datuma

angular displacement of block from zero on scalee

angular velocity of shaft (rad/s)(AI

Suffices 1, 2,3,4 refer to blocks (1), (2), (3), (4)

Note.: In general the block number is arbitrary. However theexperimental results quoted in Section 4 assume that block (1)has the largest out of balance mass and block (4) has the least.

1

1 INTRODUCTION

Shafts which revolve at high speeds must be carefully balanced if theyare not to be a source of vibration. If the shaft is only just outof balance andre.v:alv.e.s. slawly.the_vibration may merely_be anuisance but catastrophic failure can occur at high speeds even if theinbalance is small.

For example if the front wheel of a car is slightly out of balancethis may be felt as a vibration of the steering wheel. However ifthe wheel is seriously out of balance, control of the car may bedifficult and the wheel bearings and suspension will wear rapidly,especially if the frequency of vibration coincides \1ith any of thenatural frequencies of the system. These problems can be avoided ifa small mass is placed at a carefully determined point on the wheelrim.

It is even more important to ensure that the shaft and rotors ofgas turbine engines are very accurately balanced. since they mayrotate at speeds between 15 000 and 50 000 rev/min. At such speedseven slight inba1ance can cause vibration and rapid deteriorationof the bearings 1ead;ng to catastrophic fa;lure of the engine.

It is not enough to place the balancing mass such that the shaftwill remain in any stationary position, that is in static balance.When the shaft rotates, periodic centrifugal forces may be developedwhich give rise to vibration. The shaft has to be balanced bothstatically and dynamically.

Usually, shafts are balanced on a machine which tells the operatorexactly where he should either place a balancing mass or removematerial. The TM1O2 apparatus requires the student to balance a shaftby calculation or by using a graphical technique, and then to assessthe accuracy of his results by setting up and running a motor drivenshaft. The shaft is deliberately made out of balance by clampingfour blocks to it, the student being required to find the positionsof the third and/or fourth blocks necessary to statically anddynamically balance the shaft.

...,

3

2. THEORY

A shaft with masses mounted on it can be both statically and

dynamically balanced. If it is statically balanced, it will stay in

any angular position without rotating. If-it 1S aynamically balanced,

it can be rotated at any speed without vibration. It will be shown

that if a shaft is dynamically balanced it is automatically in static

balance, but the reverse is not necessarily true.

2.1 Static Balance

Figure 2.1 shows a simple situation where two masses are mounted ona shaft. If the shaft is to be statically balanced, the moment due toweight of mass (1) tending to rotate the shaft clockwise must equalthat of mass (2) trying to turn the shaft in the opposite direction.

Hence for static balance,

W2r2 . . . . 2.1W1rl =

The same principle holds if there are more than two masses mounted

on the shaft, as shown in figure 2.2.

The moments tending to turn the shaft due to the out of balance massesare:-

Directionf-bmentMass

Anti-clockwiseWlrl casal

(2) ClockwiseW2r2 CO5a2

(3) ClockwiseW3r3 casas

For static balance,

WlrlCOS al W2r2COS Q2 + W,r3COS Q, . . . . 2.2=

anti-clockwise clockwise

6

In general, values of W, r and a have to be chosen such that theshaft is in balance. However, for experiments using the TM1O2apparatus, the product Wr can be measured directly for each mass andonly the angular positions have to be determined for static balance.

If the angular positions of two of the masses are fixed, theposition of the third can be found either by trigonometry or bydrawing. The latter technique uses the idea that moments can berepresented by vectors as shown in figure 2.3(a}. The moment vectorhas a length proportional to the product Wr and is drawn parallel to

the direction of the weight from the centre of rotation.

For ~~!!£ balance the triangle of moments must close and the directionof the unknown moment is chosen accordingly. If there are more thanthree masses, the moment figure is a closed polygon as shown in.figure 2.3(b}. The order in which the vectors are drawn does notmatter, as indicated by the two examples on the figure.

If on drawing the closing vector, its direction is opposite to theassumed position of that mass, the position of the mass must bereversed for balance. For example, mass (4) shown in figure 2.3(b)must be placed in the position shown dotted to agree with the direction

of vector W..r...

Dynamic Balance

The masses are subjected to centrifugal forces when the shaft isrotating. Two conditions must be satisfied if the shaft is not to

vibrate as it rotates:-(a) there must be no out of balance centrifugal force trying to

deflect the shaft.(b) there must be no out of balance moment or couple trying to

twist the shaft..If these conditions are not fulfilled, the shaft is not dynamically

balanced.* ~: In this contezt, Msting l'efel's to rotation of the

longitudi~Z a%is of the shaft - see figul'B 2.4.

8

Applying condition (a) to the shaft shown in figure 2.4 gives:-

F2Fl =

The centrifugal force is m~2W

or 9 rw2

The angular speed of rotation is the same for each mass so that for

dynam;c balance

W2r2W.r. =

This is the same result obtained ;n equation 2.1 for the staticbalance of the shaft. Thus if a shaft is dynamically balanced it

will also be statically balanced.

The second condition is satisfied by taking moments about some

convenient datllD -such as one of the bearings.

. . . . 2.6a1 F1 a2 F2Thus. =

But from equation 2.3, F1 = Fz, so that al = az. Thus in this

simple case, dynamic balance can only be achieved if the twomasses are mounted at the same point along the shaft.

Since the reference point is immaterial, it is usually convenient totake moments about one of the masses so that the effect of that

mass is deleted from the moment equation.

Taking moments about mass (1 produces the result

0F2 (a2 - at =

Since the centrifugal force cannot be zero, al must equal a2 t as

proved above.

9

Unlike static balancing where the location of the masses along theshaft is not important, the dynamic twisting moments on the shafthave to be eliminated by placing the masses in carefully calculatedpositions. If a shaft is statically ba.:t.anced it.does not follow thatit is also dynamically balanced.

2.3 Dynamic; Balancing of Three MassesIt has been shown that dynamic balance can only be achieved for a twomass system if the masses are mounted at the same point along theshaft. It can also be shown that there are special conditons whichmust be satisfied for dynamic balancing of three masses.

Consider the case shown in figure 2.5(a). Mass (3) is positionedvertically for convenience. Condition (b) for dynamic balance canbe expressed mathematically by equating moments for centrifugalforces in both horizontal and vertical planes. In order to simplifythe equations it ;s convenient to take moments about mass (1) sothat moments due to forces on this mass are eliminated.

Horizontal:Vertical:

a2 F2 CO5a2

a2 F2 5;002

= 0

aJ FJ=

The cond;t;ons wh;ch sat;sfy equat;on 2.7 are that e;ther a2 = 0

or a2 = 90 or 2700 (;e COsa2 = 0). Subst;tut;ng these ;nto equat;on

2.8 g;ves the follow;ng condit;ons:-

(f) a2 = 0

For this condition aJ must also be zero. Thus for arbitrary

values of a2 and aJ, all three masses must be located at thesame point along the shaft.

Q2 : 90 or 2700;;For these conditions it is necessary to write down furtherequations to obtain solutions.

11

Apply;ng Condition (a) for dynamic balance:-

. . . . 2.9

. . . . 2.10Horizontal:Vertical:

Fl casal

F.

F2 CO5a2

Fl Sirnl

=

+ F2 5;002.If Q2 . 90°, equation 2.9 gives Ql = 90 or 270°.

90°, equation 2.10 reduces tOiAs swu; ng that a 1

F1 . F1 + F2

Also, equat;on 2.8 reduces to,

a2 F2 = a, F,

Combining these two equations and solving for F1 gives

-!.!.)a2Fl . F, ( . . . . 2.11

Now, if a3 is greater than a2 (as indicated in figure 2.5) it followsthat Fl must be negative, and Ql must be 270u and not 90v as assumed

above. The resulting configuration for dynamic balance is shown in

figure 2.5(b).

Thus, if the masses are distributed along the shaft, the followingconditions must be satisfied for dynamic balance:

(a)

(b)

Central mass at 1800 to other two massesMasses chosen such that.

F2 = Fl + F. . . . . 2.12

(c) Masses distributed along the shaft such that.

. . .-. 2.13- a"2-f-t .. -a~ .f)

12

2.4 Dynamic Balancing of More Than Three MassesIf there are more than three individual masses, there are no specialrestrictions which apply to the angular and shaft-~/ise distributionsof the masses and the general conditions for dynamic'balance have tobe applied to obtain solutions.

The angular positions of the masses can be determined by applyingconditions for either static balance (see section 2.1) or forCondition (b) for dynamic balance (section 2.2). The distributionof the masses along the shaft is then found by applying Condition (a)for dynamic balance. This can be done either by calculation, or bya drawing method similar to that described in section 2.1 for static

balancing.

2.4.1 Calculation techniqueThe twisting moments are resolved into components tending to twistthe shaft in the horizontal planes. The net moment in each planemust be zero if the shaft is to be dynamically balanced. As

previously noted, the equations are simplified considerably by takingmoments about the first mass. Referring to figure 2.6 the appropriateequations for a four mass system are:

~rizonta1 roments about mass (1)

0-a2F2 COSB2 + a3F3 COSQ3 . . . . 2.14+ a..F- cosa.. =

Vertical moments about mass (1)

a2F2 S;na2 + a3F3 S;na3 - a~F~ s;na- 0 . . . . 2.15=

Note that ~ , F, and F It are proportional to W ~ 2.. W,r, and W~rlt. Inthe experiment described in this manual. the values of (Wr) are knownfor the masses so these values can be used in place of those for Fin equations 2.14 and 2.15. Given the angular positions of two of themasses, the angular positions of the other two can be found by themethods given in section 2.1. If the shaft-wise locations of two of

the masses are also known, the locations of the other two can be

13.

found by substituting the known values into the simultaneousequations 2.14 and 2.15.

2.4.2 Drawing methodThe moments do not need to be resolved ;nto components in this method.Vectors represent;ng the values Wlrlal. W2rZa2. Wnrnan(proport;onal to the moments Flal. .. Fa) are drawn in the same W!yn nas for the case of static balance (;.e. moment polygon method - see

section 2.1 and f;gure 2.3). No more than two unknowns may be foundfrom the condit;on that the polygon must close. Usually two valuesof a are treated as the unknowns. Aga;n. the work ;nvolved ;sreduced if moments are taken about one of the masses.

Whichever method is used, it will often turn out that the assumedorder of the masses on the shaft will be incorrect. For example,the configuration shown in figure 2.6 may only give a dynamic balanceif the order of the masses on the shaft is altered. However, bothmethods automatically account for this since the resulting valuesof a will indicate in which order the masses must be placed on theshaft. Generally, the diagrams show purely arbitrary positions for

the masses.

1~

Fa

F-

F!

F1

F;g 2.6 Nomenclature for Arb;trary Four Mass System

')

15

EXPERIMENTAL APPARATUS3.The TM1O2 Static and Dynamic Balancing Apparatus (figure 3.1) consists

of a perfectly balanced shaft to which a set of four rectangularblocks can be clamped. The shaft runs in ball bearings and is .

driven by a 12 volt electric motor through a belt and pulleyarrangement. The motor and shaft are mounted on a metal plate whichis supported on resilient rubber feet. These feet allow the plateto vibrate when the shaft is out of balance.

D;scs hav;ng eccentric holes of differing diameters can be fitted tothe rectangular blocks so that the out of balance moment of eachblock can be varied. The blocks can be clamped to the shaft inany angular and shaft-wise positions. Two hexagon keys are prov;dedwith the apparatus. The larger on.e is used to clamp the blocks tothe shaft, whilst the smaller key fits the screws retaining the

eccentric discs.

Linear and protractor scales are provided for setting and measuringthe positions of the blocks. Adjacent to the linear scale is aslider which can be pushed up against each block in turn to readthe shaft-wise position. The slider is also used as a datum stopagainst which the blocks can be held when reading the angular

position.

The out of balance moments are measured by first fitting theremovable shaft and pulley to the main shaft. Each rectangularblock is then clamped to the shaft in turn. A cord having a weightbucket on each end is hung on the extension pulley and ball bearingsare placed in the bucket until the block has moved through 900. Theout of balance moment ;s proportional to the number of balls

required.

Micro switches actuated by a perspex dome ensure that the shaft

cannot be driven unless the dome is in position. The motor switch

on the front panel is spring loaded so that the motor cannot be left

running. The motor requires a supply of 12 Volts at 2 amps which

may be provided ei ther by one of the TecOuipment (eg E66/E90 Speed

Control) or by an ordinary automative battery.

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17

4. EXPERIMENTAL PROCEDURE AND TYPICAL RESULTS

4.1. Demonstration of Static Balance but Dynamic Imbalance

Remove the perspex dome and shaft drive belt.

Remove the discs from the four rectangular blocks using the smaller hexagon key..

Set up two of the blocks as shown in figure 4.1 (a) with a relative angular displacement of 1800 andshaft-wise displacement of 120mm. Use the slider to set and read off the positions of the blocks. Thelarger hexagon key fits the screws which clamp the rectangular blocks onto the shaft.

3.

Observe that the shaft will remain in any angular position and is therefore statically balanced4.

Connect the apparatus to a 12 volt supply. Make sure that the slider adjacent to the linear scale isclear of the blocks, then replace the shaft drive-belt and the perspex dome.

5.

Briefly run the motor. Note the severe imbalance of the shaft.6.

4.2 Simole Dvnamic Balance usina Four Masses

Remove the perspex dome and set up the four rectangular blocks as shown in figure 4.1 (b)

Prove theoretically that the shaft is both statically and dynamically balanced (see sections 2.1 and

2.3).2.

Test the shaft for static balance (see 4.1.4)3.

Replace the drive-belt and the perspex dome. Run the motor and observe the lack ofvibration, showing that the shaft is dynamically balanced.

4.

An alternative is to place the two middle blocks adjacent to one another such that their mating faces aremid-way between the two end blocks. This gives a configuration equivalent to the 3 mass system shown infigure 2.5 (b) for which F2 = 2F, = 2F3. The special condition for dynamic balance of a 3 mass system(equation 2.12) is therefore satisfied so the shaft is in balance.

* Always take the blocks off the shaft before removing or replacing the discs,

Note: The motor and rotating shaft assembly is mounted on a metal plate which may be clamped down.Ensure the plate is clamped fully down for static balancing tests. For dynamic balancing release theclamps, turn through 90° and lock down, leaving the plate free to vibrate.

18

c:::

1 2 0 ]

a) Stat;c Balance for 2 Mass System

b) Static and Dynamic Balance for 4 Mass System

Fig 4.1 Configurations for Simple Demonstrations

19

4.34.3.11.2.

3.

4.

5.

6.

1.

Static and Dynamic Balance of a Four Mass SystemEx erimenta1 determination of Wr values

Remove the perspex dome and remove the shaft drive belt.Unc1ip the extension pulley and insert it in the pulley end of

the motor dri-ven shaft.Move the apparatus to the edge of the table or bench. Loop two

or three turns of the weight bucket cord around the extensionpulley. Ensure that there are no obstructions to the

movement of the weight buckets. (See figure 4.2).Insert the eccentric disc with the small hole into one of therectangular blocks. Clamp the block to the shaft such that the

protractor scale reading is 00. Call this block (1).Gradually add the steel balls to one of the weight bucketsuntil the block has moved through 900. Whilst adding the balls,occasionally tap the shaft mountings to overcome bearing

stiction.Record the number of balls required to raise the block through900. This is proportional to the out of balance moment of the

block (Wr).Fit an eccentric disc to each block and repeat the aboveprocedure for each block in turn. Enter the results in a table

similar to Table 4.1.Remove the extension shaft and replace it in its mounting clip.8.

4.3.2 Calculation of block positions for balanceAssume that it is required to find the position of blocks (3) and(4) necessary to dynamically balance the shaft. given the positionsof blocks (1) and (2). In order to obtain a practical solution forwhich balance can be demonstrated. it is necessary to take carein choosing the positions of the first two blocks. To assist withthis. Table 4.2 gives a number of different configurations whichproduce practical solutions. The procedure is as follows:

1 Select suitable angular and shaft-wise positions for blocks(1) and (2) by referring to Table 4.2. Note that if the momentvalues are different, the configurations listed in the table can

only be taken as an approximate guide.

21

2~

3.

4

5.

6.

1

Find the angular positions of blocks (3) and (4), either bycalculation or. by drawing (see section 2 and the typical

calculations in the next section).Find the shaft-wise displacement of blocks (3) and (4), again

by calculation or drawing.Assemble the blocks on the shaft in the given and calculatedpositions using the slider to set and read the positions. Pushthe slider to one end so that it is clear of the blocks.Test whether or not the shaft is statically balanced (see

paragraph 4.1.4).Replace the drive-belt and the perspex dome, then run themotor and observe whether or not the shaft is dynamically

balanced (see paragraph 4.2.4).If the shaft is not balanced, repeat your calculations andcheck the positions of the blocks to find the mistake.When the shaft has been balanced satisfactorily, move one ofthe blocks by a small amount. Observe the effect on the

balance of the shaft.

8.

4.4 Typical CalculationsSuppose that block (1) is placed 18rnm from the end of the shaftat zero angular displacement and that block (2) is placed 100 mm from

it at 120°. The measured out of balance moments are shown in

Table 4.1.

It is first necessary to draw the moment polygon of (Wr) values, asshown in figure 4.3(a}, to determine the angular positions of blocks(3) and (4). (Wr}l and (Wr}2 are known in both magnitude anddirection. The vector (Wr}l has been drawn downwards for convenience.The lengths of vectors (Wr)3 and (Wr)~ are known and arcs can bedrawn from the ends of (Wr}l and (Wr}2 to fix the directions of the

unknown vectors.

F;gure 4.3{b} shows a graph;cal ;nterpretat;on of the angularposition of the blocks obta;ned from the moment polygon. F;gure4.3{c} shows how this is interpreted when positioning the blocks.

23

T~he shaft-wise positions of the blocks can be obtained by drawing asshown in figure 4.4. Let the unknown distance of blocks (3) and

(4) measured from block (1) be a3 and a~ , as shown in figure 4.4(b).

Figure 4.4(a) shows the angular positions of the blocks as previously

detennined. . Since b1!Jck (l} is being used as a reference, it willnot appear in the calculations.

Draw a vector (100 x 82) units long parallel to direction (Wr)2.The directions of vectors representing (Wra)3 and (Wra).. are drawnparallel to (Wr)3 and (Wr)4. Their magnitudes are found by drawingthe triangle and measuring the lengths of the sides. Note that thedirection of vector (Wra)3 must be reversed compared with thedirection (Wr)3. This means that a3 must be negative and block (3)must be placed before block 1 on the shaft.

From the vector triangle,

-12 R1TI

124 R1TI

a3 =

a.. =

To fit all the blocks on the shaft, block (3) should be positioned5 mm from the left hand end. The relative positions of the fourblocks are then:-

Angular position (°)0

120190294

Shaft-wise position (mm)

17

117

5

141

The values of a3 and a~ may be found by calculation instead of by

drawing. (See section 2.4.1).