Precision balancing any given rotor for smooth running at one or more fixed speeds requires that unbalance-originated deflections and/or forces be measured. When the rotor is balanced in situ, the standard procedure is to conduct trial runs to obtain all of the requisite information. These include at least one run with the rotor in its uncorrected state and then, for each balancing plane on the rotor, at least one run with a trial balance mass fitted to that plane. Synchronous components of deflection and/or force may be measured at numerous transducers on the machine. If the machine has $B$ short bearings, then at any one rotor speed, the dimension of the space spanned by all of the complex vectors of synchronous stator response from the trial runs should be (at most) $B$. In practice, the dimension of that space is very often larger. This paper firstly demonstrates how minimal adjustments can be applied to the (complex) measured synchronous stator vibration vectors to force those vectors into a space of dimension $B$. Standard least-square methods can then be applied to discover a suitable set of unbalance corrections. The paper shows that in virtually all cases where the noise is high, applying this procedure of projecting the data from a given rotational speed into a space of appropriate dimension before computing the least-squares calculation is beneficial. Reductions in the balancing cost function (the scalar quantity minimized by the least squares calculation) by factors of 2 to 4 are typically obtained for realistic levels of measurement noise. At very low levels of noise, the procedure is neither beneficial nor harmful. There is a strong argument that it should always be applied.

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