I spent 13 years designing electronics of this exact nature: three phase induction motor reduced voltage soft starters and variable frequency AC drives. I spent the last few of those years as a VFD applications engineer helping customers select and configure this equipment for various loads and industries around the country as well.
You will not be able to build something that is cheap and safe. The voltages and currents involved are well beyond the safety margin of a hobbyiest, especially someone who is openly avoiding buying commercial units in order to save money. Don't do it!
While the theory behind AC motor control is very straightforward, the detail level work (heat sink sizing, snubbering, gate drive requirements, de-sat protection, motor overload calculations, bus capacitor protection, etc.) can be quite tricky to get down, especially with heavy duty cycling and regenerative power modes which a carnival ride will CERTAINLY be generating. I strongly caution you against trying to build something of this nature unless you have significant experience not only in microcontrollers and embedded systems design but also significant experience in power electronics and three phase circuitry. People get hurt and killed building this stuff.
My first question for you is whether speed control is really required, or if you only require a soft start up and slow down. Do you vary the speed of the motor once it is started? If not, you may be able to get away with a MUCH cheaper reduced voltage soft starter. These units act like three phase light dimmers; they only adjust the applied voltage to the motor. You will not have a lot of torque at low speeds, but with the right design of motor (NEMA class D) you can achieve exactly what you're after with a fraction of the cost and maintenance.
If you really do need to vary the full-load speed of the motor then you are more or less stuck using a variable frequency drive. As you are aware these are expensive and if you buy cheap you are likely to replace them sooner due to your high surge current (they call this "constant torque") application. What I would definitely recommend doing if this is the case would be to contact various manufacturers (Allen-Bradley, Cutler-Hammer, SAF drives, Benshaw, Yaskawa, etc.) and ask for reconditioned units. Ask for a drive capable of delivering 150% rated current for 30s (this is usually known as heavy duty) or size the drive 30-50% larger than your nominal current rating. You will also likely be running off of generator power which is notorious for being undersized and prone to brownouts and surges as the load requirements change with the state of the equipment being run. Drives don't like that (voltage sags cause current spikes as the motor starts slipping and surges can cause you to overvoltage the bus capacitors) and have a tendency to either fault out or blow up.
I am all about the little guy building something and saving a buck, but this is not the type of project to do this on. If you really want to build a three phase AC drive, start with a little 10HP 480V motor with a hand brake on a test bench. You have all the potential for experiencing the pants-filling sensation of an H-bridge failure or a bus capacitor explosion two feet from your head but without the potential lawsuits and loss of life (except perhaps your own).
Pictures should help:
As @Asmyldof stated, the motor has no concept of hot or neutral, so the polarity of the AC connections is not important. Connect one of them to hot, and one of them to neutral. The motor will turn the same direction even if you reverse these connections. To reverse the motor, you have to remove the metal jumpers, and reposition them as indicated by the diagram.
Note that you absolutely should ground the motor housing, but this is a protective measure (the yellow/green-stripe wire is the earth/ground connection), but ground is not neutral.
Best Answer
It is a single-phase motor. Any single-phase motor can be run from a 3-phase supply of the proper voltage. Just connect it to any two phases. Are you sure that you connected capacitors of the proper microfarad value to the proper terminals? Are the capacitors rated for motor-start and motor run duty?
The diagram below shows what I think are the internal winding and starting switch connections. C2 must be the starting capacitor. It is not shown to be connected to the power connections, therefore it must be connected to an internal component that disconnects it after the motor reaches full speed after the power is switched on. Since the stating capacitor has a high capacitance, it allows a high current in the start winding. It is designed to carry the high current for only a about second or two. If it remains connected, it and the start winding will overheat and fail. C2, the start capacitor, should be the 250 uF capacitor and C1, the run capacitor, should be the 50 uF capacitor.
The reversing links change the polarity of the start winding with respect to the run winding. The connection diagram should show the links in only one position, not both at the same time. The motor must have the direction selection changed only when it is unpowered and not turning.