Faulty relay(s), capacitor or power source?
There are two distinct possibilities: 

1) The power supply is defective and cannot supply the current required by the motors.

2) Something is consuming way too much current and the supply voltage is dropping as a result of that excessive current consumption.

My first step would be to measure the current coming out of the power supply and observe what happens when the relay(s) close.  Although you can use a multimeter to do this, you may get erroneous readings.

Here's why: if this is a switch-mode power supply with fold-back current limit and something is drawing excessive current, the power supply voltage will collapse to the point where the current consumption is below the damage level.  That may also involve having the current drop to a low (sometimes very low) level.  Your multimeter will show only that low level.

Better yet would be to use a low-value shunt resistor with a scope across the resistor.  Usually, you can insert the current shunt into the power supply (-) lead, which makes connecting the scope easy.