# Balancing of separators

In this project we consider a regression problem with close connections to unbalance estimation of rotating machinery. In particular, the application of separator balancing has been an inspiration for the research.

Consider a model of a separator as shown in Figure 1. It is desired to estimate the inherent mass unbalance **x**_0 given *M* experiments . The vector **x**_*k* is a user-supplied mass which is applied in order to excite the system, *k*=1,...,*M*. The vector **y**_*k* is the measured vibrational response at a given speed of operation. The response is a function of **x**_0, **x**_*k* and the dynamic properties of the system. It is assumed that the system dynamics change in a stochastic fashion, which makes the problem hard to handle in a sound statistical way.

*Figure 1. A separator model. The stifnesses are subject to change between each new experiment. The notation* [**x**]_*i* *means element *i* of the vector* **x**.

The unknown variable **x**_0 enters the equations in a nonlinear fashion. However, the problem can be formulated as a linear regression problem, while neglecting the uncertainty due to changing system dynamics. Such estimation procedures can be found in existing literature on unbalance estimation. In this study we compare linear regression approaches with a procedure which fully takes into account the uncertainty in the experiments, which leads to a nonlinear regression problem. In addition, expressions for the accuracy of linear and nonlinear estimators are derived. Figure 2 shows the performance of three different estimators for the separator system indicated in Figure 1. The estimates A1 and A2 are based on linear regression approaches, whereas A3 is based on nonlinear regression. A1 can be found in the litterature and it is commonly reffered to as the *influence coefficient method*.

*Figure 2. Performances of three different estimation techniques.*.

## Publications

[1] Peter Nauclér and Torsten Söderström. Linear and Nonlinear Regression with Application to Unbalance Estimation, Technical Report 2008-009, Department of Information Technology, Uppsala University, 2008.

[2] Peter Nauclér. Estimation and Control of Resonant Systems with Stochastic Disturbances, Ph.D. Thesis, Department of Information Technology, Uppsala University, Uppsala, Sweden, 2008.

[3] Peter Nauclér and Torsten Söderström: Unbalance estimation using linear and nonlinear regression. Automatica, vol 46, no 11, pp 1752-1761, November 2010.