balancing - deu.edu.trkisi.deu.edu.tr/saide.sarigul/balancing.pdf · an unbalanced disk-shaft...
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BALANCING
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Static Unbalance
A disk-shaft system on rigid rails An unbalanced disk-shaft system mounted on bearings and rotated
For the unbalanced rotating system: Dynamic Bearing Reactions + Inertia Forces =0 -(FA + FB)i + mrG2i = 0
Both bearing reactions occur in the same plane and in the same direction. same direction. Balancing is performed on Balancing is performed on the plane of unbalance. After balancing no bearing reaction remains theoretically.
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Dynamic Unbalance
A long rotor mounted in bearings at A and B
• If m1=m2 and r1=r2 the rotor is statically balanced but dynamically unbalanced.
Balancing may be performed on the planes of unbalances or any other two convenient planes. After balancing, no bearing reactions remain theoretically.
Both bearing reactions occur in the same plane but in opposite directions.
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Unbalance Types
STATIC (Misalignment of axes)
DYNAMIC (Tilt of axes)
STATIC and DYNAMIC (GENERAL) (Called as DYNAMIC, in industry) (Both misalignment and tilt of axes)
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Analysis of Static Unbalance
m1R12+ m2R2
2+ m3R32+ mcRc
2=0 F=0
• mcRc is the correction quantity. It is added to the plane of unbalances.
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Analysis of Dynamic Unbalance Graphical Vectorial
Scalar
True moment polygon is 90 cw away
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Industrial Balancing Machines
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Balance Quality Grade Maximum Permissible Spesific Unbalance
MrG2=mr2
MrG=mr=u (Amont of Unbalance)
e=rG=mr/M=u/M (Spesific Unbalance) Empirically, e=constant G (Balance quality grade) (0.4-4000) For example: N=3000 rpm, G=6.3 give e=20m as max permissible eccentiricity. 1) As the rotor mass increases permissible unbalance also increases
(u=MrG ) 2) As the rotor speed increases permissible unbalance decreases
(e=constant )
For small masses and high speeds, higher quality balance grade is required.
Maximum Speed
Permissible spesific unbalance (u/M) OR Mass center’s eccentricity (e)
G. m. mr2
r
RA
rG
MrG2 M
G
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Balancing A Single-Cylinder Engine
FC
mA
F = FA,B + FC
Eliminated
Primary component reduced 50% (Altough a new vertical component added)
Secondary component does not changed
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Polar Diagram of Inertia Forces
Before adding mC:
After adding mC :
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Imaginary-Mass Approach 1) Two imaginary rotary masses are
used for each reciprocating mass.
2) Each mass is equal to half the
equivalent reciprocating mass.
3) These imaginary masses rotate
about the crank center in opposite
directions and with equal angular
velocities.
4) The total inertia force in y-direction is
zero.
5) The inertia force in x-direction is
equal to the inertia force due to
reciprocating pisto mass.
A simple model to analyse
and recover the unbalance of
reciprocating piston mass.
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Balance of Multicylinder Engines 2-Cylinder 2-Stroke Engine
First Harmonics:
Angles between imaginary masses : 360/2 = 180
S1 = 0
M1 0
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2-Cylinder 2-Stroke Engine
Second Harmonics:
Angles between imaginary masses: 720/2 = 360
S2 0
M2 0
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4-Cylinder 4-Stroke Engine
Firing Order: 1-2-4-3
First Harmonics:
Angles between imaginary masses : 720/4 = 180
S1 = 0
M1 = 0
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4-Cylinder 4-Stroke Engine
Firing Order: 1-2-4-3
Second Harmonics:
Angles between imaginary masses : 1440/4 = 360
S2 0
M2 0
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4-Cylinder 4-Stroke Engine
Firing Order: 1-2-3-4
First Harmonics:
Angles between imaginary masses : 720/4 = 180
S1 = 0
M1 0
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4-Cylinder 4-Stroke Engine
Firing Order: 1-2-3-4
Second Harmonics:
Angles between imaginary masses : 1440/4 = 360
S2 0
M2 0
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Balancing Linkages
The position of the mass center of whole linkage,
is tried to remain stationary.
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Balancing Linkages (cont’d)
rs3
rs4
rs2
Vector loop-closure eq’n:
One of the vectors, for example may be drawn from
this eq’n: e.g.
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Balancing Linkages (cont’d)
• This eq’n shows that the center of mass may be made stationary at the position, (1 and 3)
• if the following coefficients of the time-dependent terms vanish:
32 2
2 2 3 2 3 3
3
0jj r
m a e m r m a er
34 4
4 4 3 3
3
0jj r
m a e m a er
Simplification
3 3
3 3 3
j ja e r a e
32 2
2 2 3 3
3
0jj r
m a e m a er
These two
equations
must be
satisfied for
total force
balance
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Balancing Linkages (cont’d)
• These equations yield the two sets of conditions:
• and
• and
• A study of these conditions show that the mass and its location can be specified in advance for any single link (3); and then full balance can be obtained by rearranging the masses of the other two links (2 and 4).
• Without disturbing 3rd link , counterweights are added to rotating links 2 and 4. In this procedure,
•
• relations must be satisfied.
44 4 3 3
3
rm a m a
r
2 3
4 3
0 0 0 * * *
i i i i i i i i im a m a m a
Unbalanced
linkage
Counterweight
22 2 3 3
3
rm a m a
r
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Balancing Linkages (cont’d)
• Therefore gives,
• and
• If it is required to find ,
• condition must be satisfied.
* * 2 0 0 2 0 0 0( ) ( ) 2( )( ) cos(i i i i i i i i i i i im a m a m a m a m a
0 0 0* 1
0 0 0
sin sintan
cos cos
i i i i i ii
i i i i i i
m a m a
m a m a
*
im
* 0
i i im m m