Dear OULADALI ALI,
I agree with both Walt and Vibmaster. I would moreover emphasize on the possibility of multiple causes.
Based on the motor casing measurements (10.6 mm/sec or 0.42 in/sec), it seems that there is an actual vibration and it is high.
I would advise to review the pump data sheet (API forms) and see the natural frequency documented. Do the impact test and compare the results. Compare the documented natural frequencies versus measured values and measured values versus measured vibration frequency. Notice that the online spectra (System 1) are of a rough resolution and it would be better to collect the data using a higher resolution set up.
On the balance quality side, I would recommend to go to the performance sheets of the pump and motor and see the balancing and factory run test records. It is likely that the pump and motor rotors were well balanced and had acceptable vibration level. But this is the first point one should check.
Some symptoms on the balance problems would be found from vibration plots that you can extract from System1. Can you post the orbits and 1X amplitude and phase trends for example?
Again can you post a sketch/drawing to show the locations of the vibration probes/sensors?
Now, is the manufacture representative for this project aware of the high vibration problem? If he is not, somebody should communicate with him for many reasons. You might lose the warranty on the equipment if he is not officially informed about the problem.