The rotor is a body which rotates about some axis and is held by its bearing surfaces in the supports. The bearing surfaces of the rotor transmit loads to the supports via rolling or sliding bearings. The bearing surfaces are the surfaces of the trunnions or the surfaces that replace them.
Fig.1 Rotor and centrifugal forces acting on it.
In a perfectly balanced rotor, its mass is distributed symmetrically about the axis of rotation, i.e., any element of the rotor can be matched with another element located symmetrically about the axis of rotation. In a balanced rotor, the centrifugal force acting on any rotor element is balanced by the centrifugal force acting on the symmetrical element. For example, centrifugal forces F1 and F2, equal in magnitude and opposite in direction, act on elements 1 and 2 (marked green in figure 1). this is true for all symmetric rotor elements, and thus the total centrifugal force acting on the rotor is 0 and the rotor is balanced.
But if symmetry of rotor is broken (asymmetrical element is marked by red color on fig.1), then unbalanced centrifugal force F3 acts on rotor.When rotating, this force changes direction with rotation of the rotor. The dynamic load resulting from this force is transmitted to the bearings, resulting in accelerated wear and tear.
In addition, under the influence of this variable in direction force there is a cyclic deformation of supports and foundation, on which the rotor is fixed, i.e. there is vibration. In order to eliminate rotor imbalance and the accompanying vibration, balancing masses must be installed to restore symmetry to the rotor.
Rotor balancing is an operation to correct imbalance by adding balancing masses.
The task of balancing is to find the size and location (angle) of one or more balancing masses.
Types of rotors and types of imbalance.
Taking into account strength of rotor material and magnitude of centrifugal forces acting on it, rotors can be divided into two kinds – rigid rotors and flexible ones.
Rigid rotors deform insignificantly under action of centrifugal force at working modes and influence of this deformation in calculations can be neglected.
Deformation of flexible rotors can no longer be neglected. Deformation of flexible rotors complicates the solution of balancing problem and requires application of other mathematical models in comparison with the problem of balancing of rigid rotors.It should be noted that the same rotor at low speeds can behave as rigid, and at high speeds – as flexible. In the following, we will consider only the balancing of rigid rotors.
Depending on the distribution of unbalanced masses along the rotor length, two types of unbalance can be distinguished – static and dynamic (momentary). Accordingly, static and dynamic rotor balancing are referred to. Static rotor unbalance occurs without rotation of the rotor, i.e. in statics, when the rotor is reversed by gravity with its “heavy point” downwards. An example of a rotor with static unbalance is shown in Fig. 2
Fig.2 Static unbalance of the rotor.
Under the action of gravity, the “heavy point” turns downward
Dynamic unbalance occurs only when the rotor is rotating.
An example of a rotor with dynamic unbalance is shown in Fig. 3.
Fig.3 Dynamic unbalance of the rotor.
The forces Fc1 and Fc2 create a moment tending to unbalance the rotor.
In this case, the unbalanced equal masses M1 and M2 are in different planes – in different places along the length of the rotor. In static position, i.e. when the rotor does not rotate, only gravity acts on the rotor and the masses balance each other. In dynamics, when the rotor rotates, centrifugal forces Fc1 and Fc2 start acting on the masses M1 and M2. These forces are equal in magnitude and opposite in direction. However, since they are applied at different places along the length of the shaft and are not on the same line, these forces do not compensate each other. The forces Fc1 and Fc2 create a torque applied to the rotor. Therefore, this unbalance is also called moment unbalance. Accordingly, uncompensated centrifugal forces act on the bearing positions, which can greatly exceed the calculated values and reduce the service life of the bearings.
Since this type of unbalance occurs only dynamically during the rotation of the rotor, it is called dynamic unbalance. It cannot be corrected in static conditions by balancing “on knives” or similar methods. In order to eliminate dynamic unbalance, two compensating weights must be installed, which produce a moment equal in magnitude and opposite in direction to the moment arising from the masses M1 and M2. The compensating masses do not have to be set opposite and equal in magnitude to the masses M1 and M2. The main thing is that they produce a moment that fully compensates for the unbalance moment.
In general, the masses M1 and M2 may not be equal to each other, so there will be a combination of static and dynamic unbalance. It is theoretically proven that for a rigid rotor, two weights spaced apart along the length of the rotor are necessary and sufficient to eliminate its imbalance. These weights will compensate both the torque resulting from dynamic unbalance and the centrifugal force resulting from the asymmetry of the mass relative to the rotor axis (static unbalance). Typically, dynamic unbalance is characteristic of long rotors, such as shafts, and static unbalance is characteristic of narrow rotors. However, if the narrow rotor is skewed relative to the axis, or deformed (“figure eight”), then dynamic unbalance will be difficult to eliminate. (see Fig. 4), because in this case it is difficult to install correcting weights that create the necessary compensating moment.
Fig.4 Dynamic unbalance of the narrow rotor.
The forces F1 and F2 do not lie on the same line and do not compensate each other.
Due to the fact that the arm to create torque is small due to the narrow rotor, large correction weights may be required. However, this also results in an “induced imbalance” due to the deformation of the narrow rotor by centrifugal forces from the correction weights. (See for example “Methodological instructions for balancing rigid rotors (to ISO 22061-76)”. Section 10. ROTOR-SUPPORTS SYSTEM. )
This is noticeable for narrow impellers of fans, in which, in addition to force unbalance, aerodynamic unbalance is also active. And it should be understood that aerodynamic unbalance, or rather aerodynamic force is directly proportional to angular speed of the rotor, and for its compensation the centrifugal force of correcting mass, which is proportional to the square of angular speed, is used. Therefore, the balancing effect can only take place at a specific balancing frequency. At other rotational frequencies there is an additional error.
The same can be said of the electromagnetic forces in an electric motor, which are also proportional to angular velocity. So it is not possible to eliminate all causes of vibration in a machine by balancing.
Vibration of mechanisms.
Vibration is the reaction of the mechanism design to the effects of a cyclic excitatory force. This force can be of different nature.
The centrifugal force resulting from the unbalanced rotor is an uncompensated force acting on the “heavy point”. It is this force and the vibration caused by it that can be eliminated by balancing the rotor.
Interaction forces of a “geometrical” nature arising from manufacturing and assembly errors of the mating parts. These forces can, for example, arise as a result of non-roundness of shaft necks, errors in the profiles of teeth in gears, waviness of bearing raceways, misalignment of mating shafts, etc. In the case of non-circularity of the journals the shaft axis will be displaced depending on the angle of rotation of the shaft. Although this vibration also occurs at rotor speed, it is almost impossible to eliminate it by balancing.
Aerodynamic forces resulting from the rotation of the impellers of fans and other vane mechanisms. Hydrodynamic forces resulting from the rotation of impellers of hydraulic pumps, turbines, etc.
Electromagnetic forces resulting from the operation of electrical machines, e.g. asymmetric rotor windings, short-circuited windings, etc.
The magnitude of the vibration (e.g. its amplitude Av) depends not only on the excitatory force Fv acting on the mechanism with circular frequency ω, but also on the rigidity k of the mechanism, its mass m , as well as the damping coefficient C.
Various types of sensors can be used to measure vibration and balance mechanisms, including:
absolute vibration sensors designed to measure vibration acceleration (accelerometers) and vibration velocity sensors;
sensors of relative vibration – eddy-current or capacitive, designed to measure vibration displacement. In some cases (when the design of the mechanism allows it), force sensors can also be used to assess its vibration load. In particular, they are widely used to measure the vibration load of