Programm: BALANCING
Balancing Data: DATA
Rotor Balancing from a purely mathematical point of view
The Mathematics of Rotor Balancing

Abstract
All balancing weight calculations are based on the assumption of a linear relationship between the unbalance and the resulting vibration responses due to this unbalance distribution, which is the sum of an initial unbalance plus added unbalance caused by some known added weights in defined balancing planes. Practical and technical problems, not discussed here, can make it difficult apply test weights or to measure reliable vibration responses,  but once some data on different unbalance distributions and corresponding  vibration responses are available, the theory of linear equation systems provides a unique solution for any kind of selected data set. A data set consists of a collection of trial runs. Each trial run is defined by an ordinal number indicating a certain constant initial unbalance plus the total weight attached in each balancing plane and the corresponding vibration reponses measured with installed sensors at predefined rotor speeds. Balancing weights are determined by first calculating  the vibration response of selected weight sets, hereafter called (weight set) effect matrix, modified form of an influence matrix and in a second step by calculating the magnitude of these weight sets which have to be attached to minimize the measured vibrations - the so called point speed vector - for a selected trial run. A generalized approach allows to handle any special configuration as an incomplete effect matrix (less weight sets than balancing  planes) or a least square influence matrix (more data available than necessary to generate a complete effect matrix) It is shown, that the calculated effect matrix can even be used to calculate averaged responses for the trial runs. Generally the calculation of balancing weights minimizing the residual vibration responses is a least square solution as well. In certain cases this calculation even requires the solution of singular equation systems. In such cases homogenious solutions are available. Technically speaking such a solution is a self-balanced set of balancing weights, which practically does not cause any change in vibration behaviour.  Such weight sets can be used to minimize the total weight attached to the rotor.

Whenever the rotor behaviour is unknown, all systematic balancing methods start by determining the change in vibration response caused by the application of  test weights. The difference in vibration responses before and after the application of unit test weights in each balancing plane yields the influence coefficients, which form the well known influence coefficient matrix. But often the outcome of these vibration measurements can be rather inaccurate. In order to obtain a reliable results it is necessary  to evaluate all available measurements that have been collected on trial runs  with any type of balancing weight combinations in the course of the balancing process.

Let us assume, that we perform measurements on a rotor with n balancing planes and have conducted r trial runs. For each trial run we register the applied weight in each balancing plane and the measured vibration responses in p defined measurement points, that is vibration measurements in selected places at defined rotor speeds. These data can be arranged in a weight matrix U and a vibration matrix W. Thus the weight matrix U is made up of n rows and r columns, each column containing the total weight in each of the n planes, while the vibration matrix consists of p rows and r columns, each column containing  the p defined vibration responses,
the so called point speed vector. The underscore indicates that these data still include the implicit unknown initial unbalance in the weight matrix U and its effect in the vibration matrix W, which has to be eliminated, before these data can be used to calculate the effect of whatever unit weights or weight sets. This elimination can be performed by subtracting some arbitrary  trial run with its accompanying weight set from each  of the r trial runs. Considering the inaccuracy of the vibration measurements it is advisable to do some averaging right from the beginning by subtracting an average trial run with its accompanying averaged weight set. The average trial run is the sum of all trial runs divided by the number of trial runs. The averaged weight set is calculated accordingly. Moreover considering, that trial runs might have different initial unbalances this averaging has to be performed separately on each group of trial runs, which have the same presumed initial unbalance. After having eliminated the influence of initial unbalances, the results from the different groups of trial runs can be merged.

The resulting matrices of this operation are denoted as U
and W. Assuming that the rank of the matrix U equal to the number of balancing planes the well known influence coefficient matrix usually denoted as can be calculated according to the following scheme. But possibly the
matrix U is not regular. In this case the matrix U must be decomposed into an upper and a lower triangular matrix by employing the Gaussian elimination process, in order to determine the rank x of the matrix, a set of x linear independent weight sets T, a coefficient marix C and a matrix B, which represents the vibration response caused by the the unit weight sets T. The following scheme illustrates the calculation process. The balancing weight calculation

Once the classical influence coefficient matrix A is known, balancing weights for n balancing planes could be calculated by means of the least square method. The resulting balancing weights ur for a trial run wr would turn out to:

ur = [A*A] -1 A*wr

If the number and position of the available balancing planes and the selected vibration measurements meet all the well known requirements according to the n- or n+2 balancing theory, and both the matrix A plus the vibration measurement wr were accurate enough, this result would be sufficient and adequate. But as pointed out above, that in some cases the matrix A cannot be established, which means, that the matrix B has to be utilized instead. In this case, the balancing result would turn out to:

cr = [B*B] -1 B*wr    with  ur = T cr

But even this result is not general enough, because in some cases, mainly if the number of balancing planes is higher than necessary and sufficient, the matrix B*B is singular or nearly singular. In this case the solution of the equation system

[B*B] c = B*wr

yields a particular solution co which satifies the right hand side of the equation, and homogenious solutions Ch which  satisfy the equation

[B*B] Ch= 0

Thus the resulting balancing weight set would be

u = T co  +  T Ch = uo + Uh z

The weight sets Uh are self balanced, they can be added without influencing the vibration responses.

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