Order analysis on the spindle: unbalance and bearing frequencies as multiples of running speed

Anyone analysing vibration on a machining centre faces a problem that does not exist on a fixed-speed pump: the spindle changes its speed with every tool and often within a program. A spectrum in hertz therefore looks different at every speed. Order analysis solves this by referencing the frequency axis to the shaft frequency of the spindle. ‘Peak at 100 Hz’ becomes ‘peak at order 1’. This article explains what an order is, how unbalance and bearing damage differ in orders, and how our system uses order analysis on the spindle.

What an order is

An order is a multiple of the shaft frequency. The shaft frequency fr is the number of revolutions per second. It follows directly from the speed reported by the CNC control:

So order 1 means the event repeats once per revolution. Order 2: twice per revolution. An order of 6.45 means 6.45 times per revolution. That sounds abstract, but it is very intuitive. Anything rigidly connected to the shaft produces vibration at a fixed order. An unbalance in the tool holder turns with the shaft and pulls in one direction once per revolution. It therefore stays at order 1, however fast the spindle runs.

Mathematically, order analysis at constant speed is simple: you calculate the ordinary amplitude spectrum and divide the frequency axis by the shaft frequency. Our analysis library implements exactly that. The amplitudes remain unchanged; only the axis is expressed in orders. How the spectrum itself is produced is described in the article FFT, spectrum and resolution.

Why orders help with varying speeds

Suppose the same spindle runs once at 6,000 and once at 12,000 rpm. In the spectrum in hertz, the unbalance is at 100 Hz in one case and at 200 Hz in the other. An outer race defect moves in the same ratio. Comparing two measurements in hertz means comparing different things. In orders, by contrast, both measurements line up: unbalance at 1, outer race at the same non-integer order.

The ‘Order’ column does not change. That is the real benefit: a feature such as ‘amplitude at order 1’ has the same meaning at every speed. One limitation remains, however. The height of the amplitude does depend on speed, because the centrifugal force of an unbalance increases with the square of the speed. Resonances of the machine structure also amplify certain speeds more than others. So the order tells you where to look. Whether the value there is abnormal can only be judged by comparison with measurements at a similar speed.

Speed changes during the measurement

It becomes more difficult when the speed changes within a measurement window, for example during run-up. The line then smears across a frequency range in the ordinary spectrum. The classic remedy is order tracking: the time signal is resampled from equal time steps to equal angle steps of the shaft. This requires the speed profile during the measurement. After this resampling, components synchronous with rotation stay sharp even at varying speed.

Our analysis library includes this method. In day-to-day operation, however, we deliberately take a simpler and more robust route: the edge box only evaluates measurement windows in which the speed was almost constant. It reads the speed from the CNC control before and after the window and discards windows in which the speed changed too much or the spindle was running too slowly. Run-ups, braking and standstill therefore never enter the assessment.

Order 1: unbalance in tool, tool holder and spindle

Increased vibration at order 1 is the classic sign of unbalance. The mass is not distributed symmetrically about the axis of rotation, and with every revolution the centrifugal force pulls once in the same direction. On a machine tool spindle, these are the main possible causes:

• Tool and tool holder: unbalanced or poorly balanced holders, long overhang, unevenly worn cutting edges.
• Clamping and seating: a holder that does not seat cleanly in the spindle taper, or a chuck with runout.
• Contamination: chips or deposits on the taper or the holder that change the seating or act as mass themselves.
• The spindle itself: a change in the balance condition of the rotor, less common but more serious.

Important in practice: the first three causes change with the tool. Order 1 can therefore vary from tool to tool without anything being wrong with the spindle. A single outlier during a tool change is not a finding. Only when order 1 stays elevated over several measurements is it worth looking at the holder, taper and balance condition. If, on the other hand, order 1 rises slowly across all tools, this points more towards a cause in the spindle.

Order 2: twice per revolution

In vibration diagnostics, increased vibration at order 2 often points to misalignment or mechanical looseness. Loose mountings, a component with play or distortion in the assembly can cause the structure to knock twice per revolution. The assignment is less clear-cut than for order 1. Order 2 is more an indication of where to look than a diagnosis.

Our system words this with corresponding caution. If order 2 is the most conspicuous feature, the alert reads roughly ‘Vibration twice per revolution increased’, and the recommended check names alignment and mounting: check for loose parts and distortion. The alert does not claim a specific cause.

Bearing frequencies: non-integer orders

Rolling bearing damage behaves differently from unbalance. Its defect frequencies are proportional to speed, but they are not whole multiples. This is due to the kinematics: the rolling elements travel round with the cage more slowly than the shaft turns. For the assumed bearing in the table above, the outer race is at order 6.45, the inner race at 9.55, the cage at 0.40. The formulas are given in the article Detecting rolling bearing damage early.

This gives a useful rule of thumb for reading an order spectrum:

• Integer orders (1, 2, 3 …) belong to processes rigidly tied to the revolution: unbalance, alignment, looseness, the number of cutting edges on the tool.
• Non-integer orders point to rolling bearings, especially if they match the values calculated from the bearing geometry.

Two caveats apply. First, real bearings slip slightly, so the lines are not exactly at the calculated order. The search needs a tolerance band. Second, early bearing damage hardly shows in the ordinary spectrum, but it does in the envelope spectrum. Our system therefore does not look for the bearing frequencies in the order spectrum, but in the envelope spectrum. It recalculates them for every measurement window from the bearing geometry and the current speed. The principle is the same: the frequency is tied to the speed.

How we use order analysis in our system

The edge box at the machine calculates a small number of features for each valid measurement window. Two of them come from order analysis:

The edge box does not look for the amplitude at an exact point, but in a narrow band around order 1 and order 2. The width of the band depends on the frequency resolution of the spectrum. This way, a speed reading that is very slightly off does not cause the line to be narrowly missed. Only these features are transmitted, no raw data. They are joined by RMS, crest factor, kurtosis and the amplitudes at the bearing frequencies. What these features tell you is explained in the article RMS, crest factor and kurtosis.

The assessment takes place on the platform, and not against a fixed limit. Each measurement is assigned to a speed class. For each combination of machine, measurement point and speed class, the system learns its own learned baseline, i.e. the normal state of this spindle under those conditions. This solves the problem of speed-dependent amplitude: order 1 at 12,000 rpm is only compared with order 1 at a similar speed. How the learned baseline is built is described in the article Learning the baseline.

What to check when unbalance is reported

If order 1 stays clearly above the learned baseline over several consecutive measurements, our system raises an alert titled ‘Unbalance suspected (vibration once per revolution increased)’. The alert names the machine, measurement point and speed class, the measured value compared with the learned baseline and a recommended check. The recommendation reads: check tool and tool holder for unbalance, contamination and seating, check balance condition. In the workshop, that means in this order:

1. Which tool was loaded? If the deviation only occurs with one particular tool, the cause is very probably there.
2. Clean taper and holder: remove chips and deposits from the spindle taper and the face contact, check the seating.
3. Check holder and tool: balance grade, runout, overhang, condition of the cutting edges.
4. Balance condition of the spindle: if the deviation persists with different, clean and balanced tools, the spindle service provider should check the balance condition.

The urgency follows from the level. ‘Noticeable’ means keep an eye on it, no immediate action is needed. ‘Warning’ means plan an inspection at the next opportunity. ‘Alarm’ means inspect soon. The alert deliberately does not state a remaining useful life. After the check, you report back in the dashboard whether the alert was accurate. How such an alert is structured is shown in the article Alerts with a stated reason. An overview of spindle monitoring as a whole is given in the guide to spindle monitoring, and which machines we connect today is listed under CNC machining centres.

Frequently asked questions