rotors balancing v03
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4
Inertia forces system of rigid rotor
resulting moment of inertia Mi on the rigid body
Mi M x M y M z
M x dM x
V 2 xz dm yz dm V 2 yz dm xz dm V
F1G F1x
F2G F2 x F2 y
11
F
1y
Balancing of rigid rotor
system of forces equivalent to rotor inertia system
F1G F2G Fi m 2 OG F d F d d 2G 1 1G 1 F1x F2 x F d M x 1x F1 y F2 y F d M y 1y
Rotors
Rotor = part of a machine rotating at high speed about an axis, supported by bearings (machine shafts, rotating beams, fans, turbines, pumps, electricgenerators, ...) Rotor dynamic behavior is influenced by internal phenomena that rotate around the rotor axis inducing vibrations and fatigue stress loads on the rotor supports and foundation. rigid flexible rotors static and dynamic unbalance problem unbalance + critical speeds problems
V
M y dM y
V
V
moments of the elementary force of inertia dFi wrt axes x,y,z
( x 2 y 2 ) dm M y dM z
V V
5
Inertia forces system of rigid rotor
Mz O
RA
2 Fi m OG t OG n
My
Mx Fi
G
J 2 M x J xz yz J 2 M y J yz xz M y J zz
RB
Rotating force and moments of inertia determine variable (rotating) reaction forces at the supports.
system of forces equivalent to rotor inertia sys m 2 OG F d F d d 2G 1 1G 1 F1x F2 x F d M x 1x F1 y F2 y F d M y 1y
x
x
z
O
P y
dFi
xz dm 2 yz dM x dm
moments of the elementary force of inertia dFi wrt axes x,y,z
yz dm 2 xz dM y dm ( x2 y2 ) dM z dm
V
t , n = unit vectors tangential and normal to the path of centre of mass G
3
Inertia forces system of rigid rotor
y
decomposition of the force of inertia dFi on the elementary mass
J xz xz dm ;
V 2 2 V
J yz yz dm products of inertia [kgm2]
V
J zz ( x y ) dm
moment of inertia [kgm2]
6
Unbalanced rigid rotor
The rotor is unbalanced if force and moment of inertia are not null at a constant rotating speed.
RB
8
Unbalanced rigid rotor
The rotor is unbalanced if force and moment of inertia are not null at a constant rotating speed.
0 constant;
Static unbalance
7
Unbalanced rigid rotor
The rotor is unbalanced if force and moment of inertia are not null at a constant rotating speed.
Mz O
RA
Mx Fi
G
My
0 constant; Static unbalance Fi 0 Dynamic unbalance M i 0
m1 1
m2 2
14
m22 r2 F2
static and dynamic balanced rotor
m1 m2
F1 2 r1 F2 2 r2
13
Balancing of rigid rotor
Forces F1 and F2 (equivalent to the rotor unbalance) can be balanced by adding (or subtracting) two masses m1 and m2, respectively on planes 1 and 2 at radii r1 and r2
Dynamic unbalance
9
Balancing of rigid rotor
Balancing is the process of attempting to improve the mass distribution of the rotor in order to eliminate (minimize) the rotating force and moment of inertia when the rotor turns at constant speed. Balancing is usually done by adding (removing) compensating masses to the rotor at prescribed locations.
resulting moment of inertia Mi on the rigid body Mi M x M y M z components of Mi J 2 M J
x xz yz
J 2 M y J yz xz M z J zz
F1G F1x
F2G F2 x F2 y
pair of forces on balancing planes equivalent to rotor inertia system
F1
F2
F
1y
12
Balancing of rigid rotor
m12 r1 F1
Forces F1 and F2 (equivalent to the rotor unbalance and proportional to 2) can be balanced by adding (or subtracting) two masses m1 and m2, respectively on planes 1 and 2 at radii r1 and r2
2
Inertia forces system of rigid rotor
O dm
G
Fi
dFi
force of inertia dFi on the elementary mass
resulting force of inertia Fi on the rigid body 2 Fi dFi maG m OG t OG n
1
Rigid rotor
rigid rotor reference frame x,y,z rotating with the body
z
elementary mass P y
x
z-axis of rotation
angular velocity
overall volume V and mass m
F1
F2
equivalent system of forces and moments aligned on x,y axes
F
x
Fy
Mx
M y
m1 2 r1 sin 1 m2 2 r2 sin 2 Fx m1 2 r1 cos 1 m2 2 r2 cos 2 Fy m 2 r sin z m 2 r sin z M 1 1 2 2 2 2 y 1 21 2 m1 r1 cos 1 z1 m2 r2 cos 2 z2 M x
1 and 2 = balancing planes, where and/or subtract masses
Inertia forces system of rigid rotor
resulting moment of inertia Mi on the rigid body
Mi M x M y M z
M x dM x
V 2 xz dm yz dm V 2 yz dm xz dm V
F1G F1x
F2G F2 x F2 y
11
F
1y
Balancing of rigid rotor
system of forces equivalent to rotor inertia system
F1G F2G Fi m 2 OG F d F d d 2G 1 1G 1 F1x F2 x F d M x 1x F1 y F2 y F d M y 1y
Rotors
Rotor = part of a machine rotating at high speed about an axis, supported by bearings (machine shafts, rotating beams, fans, turbines, pumps, electricgenerators, ...) Rotor dynamic behavior is influenced by internal phenomena that rotate around the rotor axis inducing vibrations and fatigue stress loads on the rotor supports and foundation. rigid flexible rotors static and dynamic unbalance problem unbalance + critical speeds problems
V
M y dM y
V
V
moments of the elementary force of inertia dFi wrt axes x,y,z
( x 2 y 2 ) dm M y dM z
V V
5
Inertia forces system of rigid rotor
Mz O
RA
2 Fi m OG t OG n
My
Mx Fi
G
J 2 M x J xz yz J 2 M y J yz xz M y J zz
RB
Rotating force and moments of inertia determine variable (rotating) reaction forces at the supports.
system of forces equivalent to rotor inertia sys m 2 OG F d F d d 2G 1 1G 1 F1x F2 x F d M x 1x F1 y F2 y F d M y 1y
x
x
z
O
P y
dFi
xz dm 2 yz dM x dm
moments of the elementary force of inertia dFi wrt axes x,y,z
yz dm 2 xz dM y dm ( x2 y2 ) dM z dm
V
t , n = unit vectors tangential and normal to the path of centre of mass G
3
Inertia forces system of rigid rotor
y
decomposition of the force of inertia dFi on the elementary mass
J xz xz dm ;
V 2 2 V
J yz yz dm products of inertia [kgm2]
V
J zz ( x y ) dm
moment of inertia [kgm2]
6
Unbalanced rigid rotor
The rotor is unbalanced if force and moment of inertia are not null at a constant rotating speed.
RB
8
Unbalanced rigid rotor
The rotor is unbalanced if force and moment of inertia are not null at a constant rotating speed.
0 constant;
Static unbalance
7
Unbalanced rigid rotor
The rotor is unbalanced if force and moment of inertia are not null at a constant rotating speed.
Mz O
RA
Mx Fi
G
My
0 constant; Static unbalance Fi 0 Dynamic unbalance M i 0
m1 1
m2 2
14
m22 r2 F2
static and dynamic balanced rotor
m1 m2
F1 2 r1 F2 2 r2
13
Balancing of rigid rotor
Forces F1 and F2 (equivalent to the rotor unbalance) can be balanced by adding (or subtracting) two masses m1 and m2, respectively on planes 1 and 2 at radii r1 and r2
Dynamic unbalance
9
Balancing of rigid rotor
Balancing is the process of attempting to improve the mass distribution of the rotor in order to eliminate (minimize) the rotating force and moment of inertia when the rotor turns at constant speed. Balancing is usually done by adding (removing) compensating masses to the rotor at prescribed locations.
resulting moment of inertia Mi on the rigid body Mi M x M y M z components of Mi J 2 M J
x xz yz
J 2 M y J yz xz M z J zz
F1G F1x
F2G F2 x F2 y
pair of forces on balancing planes equivalent to rotor inertia system
F1
F2
F
1y
12
Balancing of rigid rotor
m12 r1 F1
Forces F1 and F2 (equivalent to the rotor unbalance and proportional to 2) can be balanced by adding (or subtracting) two masses m1 and m2, respectively on planes 1 and 2 at radii r1 and r2
2
Inertia forces system of rigid rotor
O dm
G
Fi
dFi
force of inertia dFi on the elementary mass
resulting force of inertia Fi on the rigid body 2 Fi dFi maG m OG t OG n
1
Rigid rotor
rigid rotor reference frame x,y,z rotating with the body
z
elementary mass P y
x
z-axis of rotation
angular velocity
overall volume V and mass m
F1
F2
equivalent system of forces and moments aligned on x,y axes
F
x
Fy
Mx
M y
m1 2 r1 sin 1 m2 2 r2 sin 2 Fx m1 2 r1 cos 1 m2 2 r2 cos 2 Fy m 2 r sin z m 2 r sin z M 1 1 2 2 2 2 y 1 21 2 m1 r1 cos 1 z1 m2 r2 cos 2 z2 M x
1 and 2 = balancing planes, where and/or subtract masses
