Here's my two cents on the subject.
As a first approximation, we consider it to resemble an increase in rotor resistance. If you look in a textbook at varying rotor resistance for wound rotor motors, you'll see the effect of changing rotor resistance on torque speed characteristic: it slides the maximum torque toward lower speeds and decreases torque performance at all speeds above the max-torque speed and increases torque at all speeds below max-torque speed.
But broken bar is a little more than an increase in rotor resistance. It creates an unbalance. We can model the unbalance as the sum of forward rotating field and backward rotating field. The backward rotating field draws current and causes voltage drop with creating useful work and in fact can create negative torque (assuming the rotor unbalance creates a stator current unbalance so we have backward rotating field on both) which decreases the net positive torque.
In the olden days it was one of the common things to look for when looking for rotor bar problems: increased starting time and oscillation on the current meter. Now we have more sophisticated predictive maintenance tools and we don't rely on increased starting time as a primary indication of broken bar or ring.
You mentioned 30 second DOL start and infrequent starting. Does the thermal damage curve really allow you to go that long?
Here's my frame of reference. We have 30 motors > 2500hp, 70 motors 300 – 1250hp, and around 1000 motors 5 – 250 hp. None of them takes 30 seconds for DOL start so by my standards it is a long start. Of course large motors are custom-designed, so in theory the rotor is matched to the task. We only have two families of motors that take more than 5 seconds to start. We have one 250 hp that starts in around 17 – 22 seconds. It is a standard motor, operated very close to manufacturer's limit. We have one 9000 hp that starts around 22 seconds (has to accelerate massive flywheel). That particular rotor has a very thick copper end ring (lots of thermal inertia as well as reduced electrical resistance) reinforced by a steel end ring for strength.
I believe for large motors this long type of start will require a very unusual rotor such as the one I mentioned. Your copper end ring doesn't look large relative to the size of the bars. There is no reinforcement.
My gut feel is that your motor is certainly underdesigned for the application as others suggested. The application includes: inertia to be accelerated, load torque during start, voltage at motor terminals during start, and frequency of starting.
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