Rotor Balancing For Multi-condition Machines

by

A.V. Barkov, M. A. Barkova, and A. G. Shablinsky
VibroAcoustical Systems and Technologies, Inc. (VAST) Saint Petersburg, Russia

Abstract

The forces that excite vibration with the rotational frequency of a machine are analyzed. Some machine conditions that occur when different forces of mechanical, electromechanical and hydrodynamical forces exceed the inertial forces of an unbalanced rotor are discussed. A software program is proposed for balancing the rotors in typical cases and when the vibration is significantly dependent on the working conditions or on the rotation speed frequency. The software includes expert programs that enable the detection of the operator’s mistakes and the machine defects that restrict the possibility of balancing the rotor.1

During dynamic balancing of the rotors of multi-condition machines during their operation, some peculiarities appear that do not occur when the balancing is made using a balancing machine. This is not only because of the increase of the balancing planes, but mainly because, in an operating machine, additional oscillating forces with rotational frequency that are not the result of the unbalance of the rotor appear. Besides this, the balancing is made using the vibration signal and not the forces applied to the supports. This means that the balancing is made using the parameters that depend on the mechanical features of the machine and its foundation. The influence of these parameters in different conditions varies and balancing for one working condition can lead to an increase in the vibration for other working conditions. For successful balancing it is necessary to take into account these peculiarities when calculating the correction weights.

An analysis of the oscillating forces in the machine shows that the vibration with the rotation frequency can be excited, not only by the centrifugal forces due to the unbalanced rotor, but also by the forces created by different sources in the operating machine. The following forces can be found most frequently:

The goal of balancing a machine in its own supports is to decrease the vibration with the rotational frequency by fixing correction weights in certain planes on the rotor. The centrifugal forces excited by these weights can also compensate for the action of some types of radial rotating forces applied on the fixed parts of the machine. As not all the forces acting in the machine have only radial rotation components, the vibration of the machine can not be suppressed entirely. Moreover, in multi-condition machine, it is possible to decrease the vibration for one operation condition efficiently, but the change in the condition can increase the machine vibration significantly. An example of such a situation is the balancing of an electric machine when the geometrical rotor axis does not coincide with its rotation axis (dynamic eccentricity of the air gap). In such machines, it is possible to decrease the vibration for the operation condition, but in this case the rotor vibration increases. The vibration of the machine and the rotor can increase when the operation conditions are changed.

Therefore, for multi-condition machine balancing, it is necessary to determine the optimal mathematical model to calculate the correction weights taking into account all the forces mentioned above. The relation between the parameters of the unbalanced weights reduced to the balancing planes (the planes where the correction weights can be affixed to the rotor) and the vibration in the measurement points is given by the following equation:

, (1)

Here is the vector of the unbalanced masses, reduced to the p balancing planes; 0 is the vector of vibration parameters, measured in n measurement points; and C is the matrix of the influence coefficients.

When there are k controlled operation conditions, then the matrix C is a rectangular one with the dimensions [p(nk)]. For the description of certain components of the forces mentioned above, equation (1) can be used if the action of these components can be substituted by the action of equivalent unbalanced masses.

The determination of the matrix C elements is the first stage of the balancing procedure. If equation (1) is linear, then it is necessary to fix a trial weight dMs ( where s is the balancing plane number) to the rotor in each balancing plane. After fixing each of these trial weights, it is necessary to measure the vector of the vibration s in n control points for each of k operation condition. In this case the elements of the matrix C will be determined by the equation:

dr x C = r- 0, (2)

where dr is the vector with one non-zero element in the number r place.

Only after the determination of the elements of the matrix C, it is possible to calculate the parameters of the correction weights from the Equation (3):

M x C - X0 = 0, (3)

where M is the vector of the correction weights.

Equation (3) can have:

For machine balancing the first or second case is most frequent, and for the multi-condition machines practically only the second case occurs. That is why to solve Equation (3) it is necessary to use the method of least squares with a functional of the following type:

(4)

Here X0r are the elements of the initial vector of displacement; Csr are the elements of the matrix of the influence coefficients; Ms are the elements of the varying vector of the correction weights.

Now, the problem of finding the correction masses can be formulated in the following way:

{F()} min. (5)

The correction masses are the components of the vector that satisfy Equation (5).

If the customer has special requirements for the vibration in different control points or for different operation conditions, special weighting coefficients can be included in the function. It will have the following form:

(6)

Here ag is the weighting coefficient for the operating condition number g from k possible conditions; br is the weighting coefficient for the control point r from the n possible control points; X0rg is the initial vibration in the control point r for the operation condition g; Csrg is the influence coefficient for the weight, fixed in the balancing plane s in the control point r for the machine operation condition g.; Ms is the varying parameter (correction weight) in the balancing plane s. The coefficients ag = 1 and br = 1 if the vibration amplitude is measured in displacement units and there are no special requirements for different control points and different machine operation conditions.

The described method of calculation correction weights was used in designing an application software for balancing the rotors of multi-condition machines (VAST-BAL). This software enables one to solve the following problems:

An important feature of the program is that it can save the results of vibration measurements and influence coefficients for each balanced machine. The knowledge of these coefficients enables the following:

The balancing should be done in several consecutive steps because of the limited accuracy of measurements and because of the possible nonlinearity of the influence coefficient matrix. One of the main aims of the VAST-BAL program is to minimize the number of these steps and the resulting labor consumption of each step. First of all, it is necessary to minimize the number of mistakes. That is why several measurements are entered for each measurement point. To increase the amount of information in the entered data, it is recommended choosing the measurement points near the machine-supporting bearings and that one make the measurements twice in radial directions at about right angles. This is done as if to substitute the point of measurement with the measurement plane. The measurements are made in the initial machine state (without trial weights) and in each trial run for all machine conditions. From the data obtained from these measurements, it is possible to detect most mistakes made during measurements and data entry into the computer. For additional control, it is possible to compare the influence coefficients of the current calculation with the data received in previous measurements or with similar machines.

The program provides the means to correct the influence coefficients. There are three reasons for necessity of this capability. First, there are often errors made in measurements of vibration parameters. To decrease this error, the program provides repeated measurements of the vibration for one measurement point and averages the results. Nevertheless, often when the vibration level becomes low, it is necessary to correct the matrix. Second, the matrix can be dependent on the value of the initial vibration vector 0. Third, non-inertial forces are present. The influence of these forces especially increases with the decrease of vibration, that is when the level of vibration approximates the required values. The manifestation of these forces can be taken into account by the change of the influence coefficient matrix. The algorithm of matrix correction includes the comparison of the calculated and measured vibration parameters after mounting the correction weights. If these parameters do not coincide the correction is done.

To decrease the labor consumption for balancing, it is necessary to minimize the number of balancing planes needed to achieve the required results. This problem is solved in the first balancing step after calculating the influence coefficients. The analysis of the influence coefficients enables one to find out the interchangeable groups of balancing planes and to exclude the duplicated planes. In this case all the stages of balancing can be made only on part of the planes. In addition the program enables the balancing by fixing the weights only on part of the planes; which part is chosen by the user.

Our experience in machine balancing shows that, for some machines, the influence coefficients differ significantly for different directions of vibration measurements. This situation appears most frequently when, in one direction of the vibration measurements, the rotational frequency is near the resonance frequencies of the nonrotating construction units of the machine. In this case,balancing by fixing the one correction weight in each balancing plane is inefficient. This situation is typical for high-speed rotation machines with light, not-rigid, foundations. In the program VAST-BAL, a special procedure is included to detect such situation and to perform the balancing.

An expert program is also included in the program VAST-BAL which enables :

The efficiency of the expert program is defined by the amount of data entered in the computer. The first version of the program, VAST-BAL, uses only the results of measurements of the amplitudes and phases of the machine vibration components with the rotation frequency and the parameters of the trial and correction weights. All the data are entered through the computer keyboard. The possibilities of the expert program can be increased significantly by using the vibration components with other frequencies. In particular, there is a possibility of identifying the type of oscillating forces that limit the balancing, and giving predictions of the development of detected defects.

Experience has shown the efficiency of the program, VAST-BAL, in balancing machines with one or several conditions in its own supports and the efficiency of the program's basic principles and algorithms.

CONCLUSION

Balancing rotors of multi-condition machines during their operation is quite different from balancing rotors with a balancing machine. This article described the reasons for this and showed how to balance multi-condition machines successfully. Software programs including expert programs to perform the balancing were described.

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