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I have a motor coupled over a synchronous belt (AT5) to a bigger tooth wheel (1:4), and the problem comes up that the belt is moving up or down (depending on the rotation direction) until it reaches the edge of the motor pulley (with flanged wheel). This leads to a noisy belt and the belt will probably be damaged over time.

The alignment of the motor axis with the other transmission axis is as good as possible and I also tried a different test stand, but I always see the same problem. So this seems to come from how the endless belts are manufactured.

I may use a phase, but I'm unsure if this is normally used and if the belt gets damaged from the phase?

So my questions is, how can I prevent the belt from walking up and down on the synchrounous pulley?

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Use a ruler or straight edge and place it across the diameter of the pulley nearest you. Check for parallel on the other pulley. If the flanges are the same width then align the pulleys.

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Figure 1. An expensive way of doing it. Image source: Mitchell Instrument.

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Figure 2. Easylaser explains why laser is better than a rule. (But they would say that, wouldn't they!)

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Use flanged backside idlers and flangeless or single flanged pulleys. From Gates - enter image description here

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When a belt runs to one side or thee other it is due to misalignment.

So, just "eyeballing" it is not good enough. You will need to make sure these are accurately positioned to have the belt running correctly.

Had to do the equivalent with a flat belt system and needed fine adjustment while it was running.

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