Hello. I would like to start a new project, a toy-claw-machine.
I am trying to understand how those claw works, and as far as I understood, they are AC (I believe), and have a coil, the more voltage you apply, the harder the grasp.
If it works with AC it should also work with DC. In either case you can use a relay. With the Arduino, you'll need a driver to drive the relay coil so if you're a beginner it's best to get a relay board with a built-in driver.
Or, there are solid state relays that can be directly driven with the Arduino, but if you go that way, get one that's appropriately rated for AC or DC. (DC & AC solid state relays are not interchangeable.)
For the power you can use a 48V [u]transformer[/u]. 48V/20 Ohms is 2.4 Amps so get one rated for 2.4 Amps or more.
AC solenoids rely on their inductive impedance XR to limit current flowing through the coil. If the coil is rated at AC and you try using it on DC you will almost certainly cause overheating, if not destruction, of the coil and drive circuit.
The advantage of an AC coil is that with the slug out of the core, the inductance and hence XR is low and the current will be correspondingly high, so creating a strong magnetic pull. Once the slug is fully pulled in, the inductance and hence XR value is higher and so the load current is lower.
Indeed AC and DC on a coil give different results. The really interesting bit in this case is that the resistance of the coil is given, presumably that is the DC resistance, but not the AC power rating. DC resistance and AC impedance of a coil are two very different things. The latter should even be specified for a frequency (50 Hz or 60 Hz matters - impedance increases with frequency).
24V and 48V operation will also give far different power rating for the same coil. I doubt that a coil that has the required force to operate the claw at 24V will survive being used at 48V for long, after all dissipated power quadruples with doubling of the voltage. Yet the advertisement suggests the coil is for 24-48V operation.