Describing the problem

Back to Basics

Balancing simply is to bring the center of mass of a rotating component to its center of rotation.

When the center of gravity does not coincide with center of rotation, centerifugal forces generated tangential to the direction of rotation causing unbalance force. table:

Symbol	Meaning
CR	Center of Rotation
CG	Center of Gravity

Where \(F_{unbalance}\) is the centerifugal force generated due to the eccentricty of CG off the CR by the amount of distance r and can be calculated from the equation:

(1)\[F_{unbalance} = m\cdot\omega^2\cdot r\]

Where:

m: Mass

\(\omega\): Rotational speed

r: Eccentricty

In equation (1), notice that \(F\alpha\left(speed\right)^2\)

Every rotating component such as impellers, discs of a motor, turbine, or compressor has a center of gravity in which the mass is distributed, and it has a center of rotation which is the line between their bearings. At the manufacturing phase, they never coincide. But why?

Simple answer: it’s too expensive to machine each component to have the same centreline of mass and rotation. Second, bearings and impellers are usually made by different manufacturers at different places. However, even though the equipment is produced by the same company, their installation setup impacts the balance and thus the center of rotation of the equipment.

### Unbalance problem Why should we be concerned about unbalanced rotors?

It generates large centrifugal forces on the rotor and bearings, resulting in high stresses on the bearings and other rotating parts of the machine. They lead to premature failure! Unplanned shutdowns happen, high-risk damages endanger lives and assets. ### Flexible Rotors To increase efficiency, larger machines are often designed with longer shafts and multiple stages, along with higher rotational speeds. As a result, machines are running above their first or second critical levels.

Failure may occur if the machine is run at a critical speed. We can all relate to the Tacoma Narrows Bridge incident.

Two measures are necessary to overcome such a problem. First, to pass the critical speed as fast as possible, and then to balance the critical mode. Otherwise, the machine will never start due to vibration protection controls.

For balancing the turbine at different critical speeds, you must be knowledgeable about the various modes and try to optimize. For example, balancing the first critical will not affect the second critical. This has been the traditional approach which is called “Modal Balancing”.

The second method is to empirically find the balancing weights which give you the best vibration at all critical and running speeds. Commonly known as the “Influence Coefficient Method”.

Least Square Model:

given:

n: balance planes(locations for corrections masses)

m: vibration readings at k different conditions of speed and load

l: different locations.

where \(m=k\cdot{l}\)

the problem is to find the optimum corrections masses in the n balance planes.

the idea here is to add a trial weight which is a mass of typically any value to be put at every n plane and measure the vibration at each m plane.

each trial mass addition round is called trial run

initial vibration matrix is:

(2)\[\begin{split}a= \left. {\begin{pmatrix} a_1 \\ a_2 \\ \vdots \\a_i\\\vdots\\ a_m \end{pmatrix}}\right\}\begin{array}{}\\\ _{\text{m measuring points}}\\{}\end{array}, a \in \mathbb{C^m}\end{split}\]

where \(\pmb{a_i}\): initial vibration measured at plane \({\pmb{i}}\) with no trial mass added

trial vibration matrix is:

(3)\[\begin{split}b = \left. {\underset{\text{n balancing planes}}{\underbrace{\begin{pmatrix} b_{11} & b_{12} & \dots b_{1j} & b_{1n}\\ b_{21} & b_{22} & \dots b_{2j} & b_{2n}\\ \vdots & \vdots & \vdots\ddots & \vdots\\ b_{i1} & b_{i2} & \dots b_{ij} & b_{in}\\ \vdots & \vdots & \vdots\ddots & \vdots\\ b_{m1} & b_{m2} & \dots b_{mj} & b_{mn} \end{pmatrix}_{}}}}\right\}\begin{array}{}\\\ _{\text{m measuring points}}\\{}\end{array}, b\in{\mathbb{C^{m\times{n}}}}\end{split}\]

where: \(\pmb{b_{ij}}\): vibration at measuring point \(\pmb{i}\) when mass were added in balancing plane \(\pmb{j}\).

trial weights matrix is:

(4)\[u= \underset{\text{n balancing planes}}{\underbrace{\begin{pmatrix} u_1 & u_2 & \dots &u_j & \dots & u_n \end{pmatrix}_{}}}, u \in \mathbb{C}\]

where: \(\pmb{u_{j}}\): mass added in balancing plane \(\pmb{j}\).

influence coefficient matrix is:

(5)\[\begin{split}\alpha= \left. {\underset{\text{n balancing planes}}{\underbrace{\begin{pmatrix} \alpha_{11} & \alpha_{12} & \dots \alpha_{1j} & \alpha_{1n}\\ \alpha_{21} & \alpha_{22} & \dots \alpha_{2j} & \alpha_{2n}\\ \vdots & \vdots & \vdots\ddots & \vdots\\ \alpha_{i1} & \alpha_{i2} & \dots \alpha_{ij} & \alpha_{in}\\ \vdots & \vdots & \vdots\ddots & \vdots\\ \alpha_{m1} & \alpha_{m2} & \dots \alpha_{mj} & \alpha_{mn} \end{pmatrix}_{}}}}\right\}\begin{array}{}\\\ _{\text{m measuring points}}\\{}\end{array}, \alpha\in{\mathbb{C^{m\times{n}}}}\end{split}\]

where: \(\pmb{\alpha_{ij}}\): influence coefficient of a mass added in balancing plane \(\pmb{j}\) has on the vibration at measuring point \(\pmb{i}\) and can be calculated as follows:

(6)\[\alpha_{ij} = \frac{b_{ij} - a_{i}}{u_j}\]

For general \(M \le N\) Least Square Equation is:

\[W = - (\alpha^T \cdot \alpha)^{-1} \cdot \alpha^T \cdot A\]

but as \(\alpha \in \mathbb{C}\), we should replace \(\alpha^T\) with \(\alpha^H\) where:

\(\alpha^H\) is the Hermitian transpose or the conjugate transpose of a complex matrix, can be expressed as:

\(\alpha^H=(\bar{\alpha})^T\), where \(\bar{\alpha}\) : the conjugate of matrix \(\alpha\).

rewriting the equation:

(7)\[W = - (\alpha^H \cdot \alpha)^{-1} \cdot \alpha^H \cdot A\]

### The Mathematical Model Balance of flexible rotors is important in order to get optimal vibration levels at all rotor bearings since balancing weights must be calculated for each balancing plane. Turbines and compressors usually have measuring planes that are more than balancing planes. This creates an over-determined mathematical model that needs optimization methods to get the best results. The optimization problem is set to be convex optimization with constraints regarding balancing weights and maximum vibration allowed for certain locations. The challenge was also to beat the problem of ill-conditioned planes multicollinearity The whole work was a trial to convert Darlow “Balancing of High-Speed Machinery” work published 1989 to a working python script that can be used in the filed.