I am confused about the above question. The whole point of a generator is to produce a current (by inducing a current) in a wire by moving wire in the two poles of a magnet right? So with this is mind, I am wondering why I only see examples of U- shaped armature that is coiled with wires rotating between the two poles of the magnet. Why can't you also induce current by the U-shaped armature simply moving up and down between the two magnetic poles? The wire is still moving perpendicular to the magnetic field so the EMF should also be induced in this case? So why not have the armature move up and down as well? Thanks.
If you research the web you'll find that the word armature in the area of electrical machines was selected to name the part of the machine that moves. This in contrast to the name stator, the part which remained stationary.
But these terms were more widely used before the invention of the brushless (DC) motor where the moving component is not comprised of coils but rather a series of permanent magnet poles. And the commutation, rather than brushes is done using solid state switches that create a moving field in the stationary coil.
And yes from pure physical principles the machines all work from the creation of electromotive force. So it can be coil pushing coil, magnet pushing coil or coil pushing magnet. The naming, today is less significant, and more important the imagination to consider the basic principles when it comes to design. Don't get stuck on names!
We generally use the generators to convert mechanical energy to electrical energy. As far as the sources from which we generate electricity are concerned, It's easier to derive electrical energy from the rotation of a shaft rather than an up-and-down motion.
However, there are so called Linear Alternators which essentially does that. A linear alternator is most commonly used to convert back-and-forth motion directly into electrical energy. This short-cut eliminates the need for a crank or linkage that would otherwise be required to convert a reciprocating motion to a rotary motion in order to be compatible with a rotary generator.
But they are not used at powerhouses :) They can be found at devices like "Faraday flashlights". They contain a coil and a permanent magnet. When the appliance is shaken back and forth, the magnet oscillates through the coil and induces an electric current. This current is used to charge a capacitor, thus storing energy for later use. The appliance can then produce light, usually from a light-emitting diode, until the capacitor is discharged. It can then be re-charged by further shaking.