Chapter 3 Rigid rotor balancing

Transcription

Chapter 3
Rigid rotor balancing
By Danmei Xie
The main purposes of this course
increase understanding of
rotor vibration phenomena
provide a means for controlling
or eliminating these vibrations
Basic theory of vibration
Reasons and features of vibration
Methods of rotor balancing
Wuhan University- Dr. Danmei Xie
2012/9/25
Lateral vibration
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
Rigid rotor

a rotor which operates substantially below its first
bending critical speed.
A rigid rotor can be brought into, and will remain in,
a state of satisfactory balance at all operating speeds
when balanced on any two arbitrarily selected
correction planes//

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Unbalance
 Static unbalance
 Dynamic unbalance
 Combined unbalance

e1
e2

F1

F2

 F2
F2

F
e1
e2

F1
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3.1 Two terms & conditions of rigid rotor balancing
For a single degree-of-freedom forced vibration system, if the
damping is given, then the amplitude and the phase of the system
under forced vibration should be expressed as followings
A

Fc

4 
K (1  2 ) 
2
n
n
Where, 
2
2
2
2
  arctan 2 2
n  
c
is the coefficient of resistance
m
c is the coefficient of damping
K is the coefficient of stiffness
Fc
 y st is the static displacement//
k
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A
ma 2
(k  m 2 ) 2  b 2 2

a( / n ) 2
(1   2 / n2 ) 2  4( / n ) 2
 2 /  n 
b

  arctan
 arctan
2
2
2 
k  m
1  /n 
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Loose or soft bearings
tight or hard bearings
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a) Parallel Eccentricity
b) Conical Eccentricity
c) Self-Canceling Eccentricity
d) Total Eccentricity
Figure 4.3 Distribution of Mass Centroidal Axis Eccentricity and the Effective Components in Terms of
Rigid Rotor Response
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For rigid rotor, according to the kind of
unbalance, balancing
 Static balancing refers to single-plane balancing
 Dynamic balancing refers to two-plane balancing,
subdivided as
 low speed balancing
 and high speed balancing//

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shaft
rail
Parallel rail
roller
Static balancing rig
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低速平衡台示意图
Low speed
balancing rig
(a)摆动式平衡台；(b)弹性体式平衡台
1－轴瓦座；2－轴瓦；3－千分表挡板；4－紧固螺栓；
5－弧形承力座；6－承力板；7－台架
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静平衡台及轨道截面形状
1－轨道；2－台架
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3.2 Methods of rigid rotor balancing
3.2.1 Two trial runs (steps) (low speed balancing)
 Procedure





Measure the initial rotor vibration A0 at a speed firstly (e.g.
balance speed), as uncorrected rotor data
Install a trial mass P at a position (usu. zero position ), and
measure rotor vibration A1 at the same speed
Shift the trial mass to the second position (e.g. 1800) , and
measure rotor vibration A2 at the same speed
Draw a geometric figure//
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Make △OMD，OM:OD:DM=A0:A1/2:A2/2 M
prolong MD to MC, and make MD=CD
Ap1
A2
prolong OD to ON, and make OD=DN
N
A0
Link OC and MN
Vector analysis
 A0←Fc
 A1 ← Fc ＋P → A1 = A0 ＋Ap1
 A2 ← Fc －P → A1 = A0 ＋Ap2
Make a circle, its radius is OC
D
A1
O
β Ap2
C
Measure the angle∠SOC
S
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
Calculate the correction mass Q：
A0
Q  P
Ap1
1) How to choose the suitable trial mass P?
 For rotors balanced on balance rig :
r is the correction mass radius
k is the coefficient of sensitivity
 For rotors balanced on bearings :
P  A0 / kr
P  60  A0 / r
A0 Mg
P
r 2 s
s is the sensitivity
2) On where should the correction mass or trial mass be placed?
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Mass groove
Correction mass
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3.2.2 Three trial runs (steps)

Procedure
 Measure initial rotor vibration A0 at a speed
 Install a trial mass P at a position (zero position), and
measure the rotor vibration A1 at the same speed
 Shift the trial mass to the second position (1200) , and
 Shift the trial mass to the third position (2400) , and
Plot a geometric figure
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3
2
C
1
A0
O
A1
A2
A3
Plot three semicircles with the radius of A1,A2,A3 respectively
Find three points 1,2,3 on the semicircles (equilateral triangle△123)
Find the geometrical point C in the triangle, make a circle，on
which point 1, 2, and 3 pass simultaneously
O1= A1 , O2= A2, O3= A3, OC= A0,
Then C1= Ap1, C2= Ap2, C3= Ap3,
A0
Q  P
Measure the angle ∠OC1
Wuhan University- Dr. Danmei A
Xie
p1
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3.2.3 The polar plot method



Measure the initial rotor vibration A0 at first,
referred to as uncorrected rotor data.
Install a trial mass of known size at a
predetermined angular location，then measure
the rotor vibration A1 at the same speed, referred
to as trial mass data.
Use this rotor vibration data and fairly simple polar
plotting techniques, calculate the appropriate
unbalance compensation, or correction mass
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subtraction of vector
Ap
trial mass data
A1
Ap
A0 uncorrected rotor
β
A0
Q  P
Ap1
-A0
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Example 1






Known A0∠α0=275μm ∠40, ( single balance plane)
Calculate trial mass: P=60×275/R=658≈670 g
Add the trial mass on the rotor, measure A1∠α1=
290μm ∠80
Calculate Ap∠αp= 216.6μm ∠149
Measure ∠β＝71.1
Then Q=P× A0 / Ap＝ 670 × 275 / 216.6=850.7 g
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Û
Û
the uncorrected rotor data
Tˆ
the trial mass data
D̂
the subtraction of vector Û from vector T
ˆ
Figure 4.4 Illustration of Polar Plot Calculation Procedure for Single
Plane Rotor Balancing
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



Application of single plane balance
Rotor of fans, and pumps etc
Couplings of steam turbine
Shaft of main oil pump for ST
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Sample: Machine to be balanced
Impeller Parameter:
• Diameter: 1400mm
• Thickness: 500mm
Bearing1
Bearing2
• Blade Number: 12
• Material:Fiberglass-Reinforced
Impeller
Plastics
• RPM: 1825 r/min
• Bearing Model: ?
Motor Parameter:
• Power: 75kW
• RPM: 1500 r/min
Motor
Others:
• Belt transmission
• Spring base
• Manufacture: LG
3.4 Influence Coefficient Balancing


Influence Coefficient - A complex value representing
the effect of the addition of a unit trial mass in a
specific balancing plane on the rotor response at a
particular measurement plane.
Influence Coefficient Balancing - An entirely empirical,
flexible rotor balancing method which uses known trial
masses to experimentally determine the sensitivity of a
rotor; and subsequently uses this sensitivity information
to determine a set of discrete correction masses that
will minimize synchronous vibrational amplitudes
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
αij are the influence coefficients relating the rotor response

for the specified sensors and speeds to the balancing
In the simplest case, a single trial mass is used for each plane,
one plane at a time, and
 ij 
xij  xi 0
Tj
where xi0 is the ith vibration reading with no trial masses
installed,
xij is the i th vibration reading with a trial mass installed in
the j th balancing plane,
and T j is a complex value representing the amplitude and
angular location, in rotating coordinates, of this trial mass .
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Example 2




Known A0∠α0=275μm ∠40, ( single balance plane)
Calculate trial mass: P=60×275/R=658≈670 g
Add the trial mass on the rotor, measure A1∠α1= 290μm
∠80
Calculate Ap∠αp= 216.6μm ∠149
Calculate influence coefficientαij
216.6149

 323.3149 ( m ) / kg
0.670


 A0    Q


 A0  27540  275220  323.3149  Q

Q

 A0


275220
 0.851(kg)71
323.3149
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Basic Principle
Of 1 Plane Balancing
1 select a plane to fix trial mass and a point
to measure, draw scale of phase and sign of
0o phase
2 measure initial vibration A0(phase and
amplitude)
3 fix a trial mass Q on the plane, measure
vibration A1
4 calculate influence coefficients:
amplitude
phase
RPM
5 calculate balancing mass P:
 
A1  A0
Q
P   A0
www.sendig.
36com
2012/9/25

Using simple statics, but with complex valued forces, we have
 F  RL  U1r1 2  U 2 r2 2  RR  0
( M ) L  RL LL  U1r1 2l1  U 2 r2 2l2  RRlR  0
U1  [ RR (lR  l2 )  RL (l2  lL )] /[ r1 2 (l2  l1 )]
U 2  [ RL (l1  lL )  RR (lR  l1 )] /[ r2 2 (l2  l1 )]
where
∑F represents the sum of the forces on the rotor
(∑M )L represents the sum of the moments about the left end of the rotor
RL, RR are the bearing reactions at the left and right rotor supports,
respectively
U1, U2 are the unknown equivalent discrete unbalances at axial locations 1
and 2 in Figure 5.
r1, r2 are the corresponding radii for application of the correction masses
ω is the speed of rotation in radians per second//
34
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B
A
T
brg
GEN
brg brg
brg
＃2
＃1
• For two balance planes


 Measure the amplitudes of the two bearings A0、B0

 Add the trial mass Pa on the A end of the rotor, 
measure the amplitudes of the two bearings A01、B01

 Add the trial mass Pb on the B end of the rotor,
measure the amplitudes of the two

 bearings
Calculate influence coefficients//A02、B02
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



Amplitude caused

 bytrial mass Pa
At A end
At B end
A1  A01  A0



B1  B01  B0

Amplitude caused by trial mass Pb



 At A end
A2  A02  A0



 At B end
B2  B02  B0
influence
 At A end
 At B end
influence
 At A end
 At B end
A1
A2

coefficient caused by trial mass Pa
 
a
1  A1 / Pa
 
b
1  B1 / Pa

coefficient caused by trial mass Pb
 
a
 2  A2 / Pb
 
b
 b  B2 / Pb
A0
A01
A02
B1
B01
B02
B2
B0
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
Calculate the correction masses
At A end
 b  a
A0 2  B0 2
Qa   b  a  a  b
1  2  1  2
At B end
 a  b
B01  A01
Qb   b  a  a  b
 1  2  1  2
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


3.3 Features of rigid rotor balancing
no more than two balancing planes are required
for complete balancing of a rigid rotor
Their balance can be accomplished at any speed,
i.e. it is not related with balance speed//
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





Typically, the balancing speed for hard-bearing machines is
between 600 and 1800 rpm
Generally, higher speeds are used for lighter rotors or where tight
balancing tolerances are encountered
The bearing forces, and thus the sensitivity to unbalance, is
proportional to the square of the speed of rotation
Therefore, when more sensitivity is required, higher balancing
speeds can be used
The balancing speed is, however, limited by the flexibility of the
rotor and supports in that it must remain well below the lowest
rotor critical speed
Sometimes, particularly in the case of very light rotors, it is not
possible to attain sufficient sensitivity at allowable balancing
speeds. In this case, it may be necessary to use a soft-bearing
balancing machine//
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





History of rotor balance
Jeffcott demonstrated the necessity of rotor balancing in his
classic paper in 1919
The first significant contributions to the rotor balancing
literature did not appear until about 1930
Prior to the 1950s, the balancing literature was concerned with
the balancing of rigid rotors and, in a few cases, very simple
flexible ones
The first flexible rotors of significance to be built were steam
turbine rotors
Initially, these rotors were balanced using simple, rigid-rotor
procedures//
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
problems associated with use of rigid rotor balancing
machines

These problems are generally a result of improper or
inappropriate use of rigid rotor machines and can be avoided if
the supervisory engineer is aware of them
It is essential that rigid rotor machines not be used for
balancing rotors which are, in fact, flexible
Rotors should have the same centers of rotation on the
balancing machine as they do in operation. For example, if a
rotor is to be supported by rolling-element bearings, it should,
if possible be balanced mounted in these same bearings. If the
same center of rotation is not used, substantial unbalance can
be introduced which may have a detrimental effect on bearing
and rotor life
A third source of potential problems with rigid rotor
balancing machines occurs when a rotor stack-up must be
disassembled after balancing in order to be installed//
①
②
③
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




Flexible rotor balancing procedures can generally be divided
into two groups
modal balancing, in 1953 by Grobel-a trial mass procedure
and influence coefficient balancing, in the early 1960s
As very limited instrumentation and computational tools were
available at that time, a balancing method was needed that did
not depend heavily on such tools
Modal balancing fit naturally into these requirements as only
simple calculations are required and operator insight is the
primary ingredient, rather than large quantities, and quality, of
vibration data//
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



Most mechanical engineering handbooks and general
references include some mention of rotor balancing. In most
cases, this mention is strictly limited to rigid rotor balancing.
A number of technical papers concerned with general and
rigid rotor balancing have also been published
More recently, similar discussions were presented in both
editions of The Shock and Vibration Handbook, in 1961 and
1976.
While the two editions presented slightly different
discussions, the content was basically the same
The primary emphasis was on an updated review of machines
and methods for balancing rigid rotors//
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



Influence coefficient balancing was developed some years
later, made possible by improvements in instrumentation and
the introduction of the digital computer
Consequently, the use of large quantities of high quality data
was substituted for operator insight as the central component
in the balancing Procedure
Subsequently, the Unified Balancing Approach was
developed as an empirical method, in the mold of influence
coefficient balancing
It was designed to take advantage of the modal nature of rotor
response, so as to avoid some of the difficulties of influence
coefficient balancing//
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




In the early 1940s, Kroon published two papers on rotor
balancing, which were apparently intended as a design guide
In the first of these papers, Kroon described the theory behind
synchronous rotor vibration and the need for balancing of both
rigid and flexible rotors
In the second paper, he discussed a number of specific rotor
balancing machines and methods
While this discussion was primarily concerned with rigid rotor
balancing, a graphical method was described for two plane
balancing of flexible rotors
He also presented a brief, practically oriented discussion of
field balancing//
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




A number of other papers concerned with general rotor
balancing have been published, including papers by Muster
and Flores, Jackson, and Stadelbauer
Muster and Flores compiled rigid rotor balancing criteria from
a variety of sources and compared these criteria with the
actual criteria used in American industry at the time (1969)
Jackson described a procedure for single plane field
balancing of rigid or flexible rotors using an oscilloscope
lissajous pattern of the rotor orbit
Van de Vegte and Lake proposed a procedure for balancing
rigid rotors during operation which utilizes actively
controllable, eccentric disks
They indicated the potential adaptation of such a mechanism
to modal balancing of flexible rotors, but provided no details//
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




Bishop proposed to use this same balancing head design for
balancing flexible rotors
This would be done with a single head located, axially, as far
as possible from all mode shape nodes
Then, the head would be readjusted, using a simplified
procedure also proposed by Bishop, in the vicinity of each
critical speed during each run up and run down of the rotor
Gosiewski also promoted the use of balancing heads for
balancing flexible rotors
Unlike Bishop, he proposed using multiple heads in a
procedure which might be described as automatic, and
continuous, influence coefficient balancing//
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

He defined the procedure and control algorithm for
implementing this approach using on-line computer control
Gosiewski also evaluated several balancing head designs,
including that of Van de Vegte and Lake, and concluded that
cartesian adjustment approaches, which use laterally
adjustable orthogonal masses, are superior to polar adjustment
approaches, which use rotating eccentric masses//
2012/9/25
Summary


Rigid rotor: the rotor being balanced does not elastically
deform at any speed up to its maximum design speed
Two terms & conditions of rigid rotor balancing
Fc
A
K (1 

4 
)

2
2
n
n
2
2
2
2
  arctan 2 2
n  
two- or three-steps method
Influence Coefficient Balancing
Features of rigid rotor balancing
no more than two balancing planes are required for complete
balancing of a rigid rotor.
Their balance can be accomplished at any speed, i.e. it is not
related with balance speed//
2012/9/25
Questions







Can you distinguish between the trial mass and the correction
mass in rotor balancing ?
Please list the procedures of three steps balancing of rigid rotor?
Please explain the influence coefficient. Can you balance a
rigid rotor by using influence coefficient balancing? And how?
The rigid rotor balancing, is based on two important
assumptions, what are they?
Please explain the least-squares method used in rotor balance?
Please explain why a rigid rotor can always be balanced in two
planes?
Try to describe the development of rigid rotor balance (from a
single trial mass run, two trial mass runs, three trial mass runs,
polar plot, influence coefficient method ) //
2012/9/25

Chapter 3 Rigid rotor balancing

Transcription

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