Most machines have vibration problems due to the natural unbalance. The unbalance may be due to faulty design or poor manufacture and assembly. The following is a list of sources of unbalance in rotating machinery.
Dissymmetry (core shifts in castings, rough surfaces on forgings, unsymmetrical configurations)
Non homogeneous material (blowholes in cast rotors, inclusions in rolled or forged materials, slag inclusions or variations in crystalline structure caused by variations in the density of the material)
Distortion at service speed (blower blades in built-up designs)
Eccentricity (journals not concentric or circular, matching holes in built-up rotors not circular)
Misalignment of bearings
Shifting of parts due to plastic deformation of rotor parts (windings in electric armatures)
Hydraulic or aerodynamic unbalance (cavitation or turbulence)
Thermal gradients (steam-turbine rotors, hollow rotors such as paper mill rolls)
When a rotor is balanced statically (single plane), the shaft axis and principal inertia axis do not Coincide. Single-plane balancing ensures that the axes have only one common point, usually the center-of-gravity. Static unbalance is often found in rotors which have a low length to diameter ratio such as in ventilators and car tyres.
In the case of Dynamic Unbalance, the central principal axis is neither parallel to nor intersects the shaft axis. Dynamic unbalance is usually found in rotors with high length to diameter ratio such as in turbines and multi-rotor assemblies.
There exist two types of dynamic imbalance: the two plane imbalance in rigid rotors, where the angular velocity is smaller than the critical angular velocity; and Flexible rotor imbalance, where the angular velocity is larger than the critical angular velocity, where the shaft deflects variably as speed increases. This means that at certain driving frequencies due to the driving angular velocities there exists resonance with certain particular parts in the product being tested - due to a match with this part's natural frequency. This can be plotted onto a graph and a spike is clearly indicative of these critical frequencies. These critical speeds are important since rigid rotors are usually below 70% of this value whilst flexible rotors are above 70% of this value.
Before proceeding to the balancing of a product it is best to first determine whether this has a single or two plane unbalance. A particular guide states that: single plane is to be used if the length to diameter ratio is less than 0.5 at less than 1000 RPM, or if the length to diameter ratio is larger than 0.5 at less than 150 RPM;
two plane balancing is to be used if the length to diameter ratio is less than 0.5 at above than 1000 RPM, or if the length to diameter ratio is larger than 0.5 at more than 150 RPM. Having said that, it should be kept in mind that this is just a guide and is not a reliable and true source in all cases.
In order to have perfect balance the principal axis must be rotating about the center-of-gravity in the longitudinal plane characterized by the shaft axis and the principal axis. In other words, balancing is the process of alignment of the principal axis of inertia with the axis of rotation by the removal, addition or adjustment of mass to the rotor. High speed rotors are balanced by removing mass, whilst large rotors usually have readily manufactured holes to which balancing masses can be attached.
In static unbalance, it is enough to add or remove mass in the single plane and its balancing can be performed without spinning the rotor (gravity principle) eventhough it is usually measured and corrected by spinning the rotor. In the case of dynamic unbalance it is required to add or remove masses in two planes. It follows that measurement and correction of the unbalance is only possible via spinning the rotor, since the rotor may have dynamic unbalance and at the same time have no static unbalance/ ecentricity. Theoretically it is not important which two radial planes (to which the masses are added) are selected since the same rotator effect can be achieved with appropriate moments (composed of both mass and eccentricity), irrespective of the axial location of the two planes. Having said that, in practice, the choice of planes may be important. Usually, it is best to select planes which are separated axially by the furthest distance possible in order to minimize the magnitude of the masses required. The rotor to be balanced should be easily accessible, and should have provision for mounting trial masses at various angles around it. The mounting points should preferably be at the same radius from the axis of rotation to simplify calculation. The following is a manual procedure of how to statically and/or dynamically balance a machine's rotor.
Measurement of the intial unbalance in two planes (amplitude and phase). Phase is measured with respect to some reference point at the rotor diameter, usually marked by reflective tape or silver paint marker.
The trial weight is placed in one plane and the unbalance is measured.
The trial weight is placed in second plane, and the unbalance is measured.
The trial weight is placed in the second plane, and unbalance is measured.
The correct value of weights and their positions is calculated.
In industrial applications there exist automatic balancing machines which can detect the imbalance, state where and which mass is required for balance and sometimes even remove or add weight accordingly. Basically these machines have spring supported bearings at each end of the rotor. This is done so that the unbalanced force is detected via its motion. This is commonly referred to as the cradle balancing machine and is better described in the figurexxx.
In dynamic balancing it would be better to theoretically split a long rotor into a series of thin disks each having some unbalance as can be seen in figure xxx.
Unbalance is represented by the mass 'm' having eccentricity 'e' which rotates with angular velocity ''.
This mass generates a centrifugal force of at an amplitude of ' ' and rotational frequency' ' which is equal to the rotating speed. The resulting equation of motion is :
Where 'M' is the mass of both the rotating and stationary components. This equation is identical to the differential equation of motion of forced harmonic vibration with'' substituting ''.
Hence the steady state solution can be replaced by :
The reduced non dimensional equation for the amplitude is:
The reduced non dimensional equation for the phase angle (the angle between the base ecitation and the response) is:
The complete solutions are:
The graph in figurexxx shows the response of an unbalanced rotor as a function of the frequency in terms of phase angle and . This shows that at low speeds, where r is small, the amplitude of the motion of
the mass (M - m) is nearly 0, whilst at very large values of r, the amplitude becomes constant, at a value equal to . At resonance, when r = 1, the amplitude is reduced if there is damping in the system and the phase angle φ is 90° thus the response lags the excitation by 90°.
For a system to be dynamically balanced it should have a zero summation of both the net centrifugal force and mass. Assuming that some kind of weight unbalances exist in the two-plane system, the following traditional two-plane vector equations for the initial unbalance response of a linear mechanical system can be used:
http://zone.ni.com/cms/images/devzone/tut/a/63059f241215.gifInitial Vibration Vector at Bearing 1 (Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241216.gifInitial Vibration Vector at Bearing 2 (Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241217.gifSensitivity Vector at Bearing 1 to Weight at Plane 1 (Grams/Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241218.gifSensitivity Vector at Bearing 1 to Weight at Plane 2 (Grams/Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241219.gifSensitivity Vector at Bearing 2 to Weight at Plane 1 (Grams/Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241220.gifSensitivity Vector at Bearing 2 to Weight at Plane 2 (Grams/Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241221.gifMass Unbalance Vector at Plane 1 (Grams at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241222.gifMass Unbalance Vector at Plane 2 (Grams at Degrees)
In order to calculate the influence coefficients of the system some trial weight must be added at both planes to acquire vibration vectors under different conditions. For a linear system, the addition (or removal) of a calibration weight W1 at plane 1 should vectorially sum with the existing unbalance U1 to produce the following new pair of vector equations:
http://zone.ni.com/cms/images/devzone/tut/a/63059f241225.gifVibration Vector at Bearing 1 with Weight W1 at Plane 1 (Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241226.gifVibration Vector at Bearing 2 with Weight W1 at Plane 1 (Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241227.gifCalibration Weight Vector at Plane 1 (Grams at Degrees)
Removal of the calibration weight at balance plane 1, together with another calibration weight W2 at balance plane 2 produces the following pair of vector equations:
http://zone.ni.com/cms/images/devzone/tut/a/63059f241230.gif Vibration Vector at Bearing 1 with Weight W2 at Plane 2 (Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241231.gif Vibration Vector at Bearing 2 with Weight W2 at Plane 2 (Grs,p-p at Degrees)
http://zone.ni.com/cms/images/devzone/tut/a/63059f241232.gif Calibration Weight Vector at Plane 2 (Grams at Degrees)
The equations above contain eight known vector quantities: six vibration vectors and two calibration weights. The calculation initially solves for the four unknown balance sensitivity vectors, and then for the two mass unbalance vectors. The following expression provides a general solution for balance sensitivity vectors for the solution:
The subscript 'm' specifies the measurement plane, and the subscript 'p' identifies the weight correction plane. Combining the solutions for the four balance sensitivity vectors within the initial equations results in the following mass unbalance at both correction planes:
The calculated mass unbalance vectors (http://zone.ni.com/cms/images/devzone/tut/a/63059f241236.gif and http://zone.ni.com/cms/images/devzone/tut/a/63059f241237.gif) represent the amount of weight that should be used at each balance correction plane. The angles associated with these unbalance vectors represent the angular location of the mass unbalance. This means that the weight can either be removed at the calculated angles, or else added at the opposite side of the rotor.
Modern balancing machines perform such calculations automatically and in many industrial applications the rotors must be balanced periodically. For such demands, portable balancing equipment has been invented.