Replace a battery with a power supply
Replacing a battery from a battery-operated equipment with a power supply can be tricky. Especially when the equipment uses an electric motor. The problem is that an electric motor can draw very large startup current - it can be as 10-20 times the nominal for a couple of seconds. A battery is able to supply that current, while a typical power supply with overcurrent protecton may latch or enter hiccup mode, from which it cant escape.

First of all you need to determine the nominal motor current and the maximum current. The maximum current flows when the motor is mechanically overloaded. In your case it is when you try to cut bushes or some really tough grass with your mower.

There are a couple of approaches one might try.


Put a very heavy low frequency transformer rated for the nominal current, put a diode bridge and a very large capacitor rated for the nominal current. Assume the transformer will whitstand the maximum current. In most cases it will. The drawbacks are weight, size &amp; price. It may be heavier than your original battery.
Put a SMPS (switch mode power supply) that can withstand the maximum motor current - it will be expensive. The power supply must be enclosed this way so no water can reach it.
Put a SMPS that is designed to work with a battery and also put a battery smaller than original. - Also expensive, also must be waterproof.
Forget about all the crap and get a new battery - it is the easiest and (in the end) cheapest way.


P.S. One more thing - 


  The mower is an outdoor equipment which is supposed to use in a highly
  moisture environment. When you buy equipment from the market it has
  certificates (UL, CE) that declare its compliance with safety standards. When a 12-24V appliance
  is being designed high voltage hazard is not in the scope and no steps
  are taken for protection. When you wire up this equipment to 110V/220V
  you must really know what you are doing.