If the engine is vibrating within a certain rpm range, what is the likely cause?

If the entire engine vibrates in a certain frequency range, then there may be a resonance between the exciting frequency which is the engine rpm/60 and one of the six natural frequencies of the engine on its mounts.

What do you mean by a "resonance"?

Oh, hummm

Well first of all google the words "Natural frequency" and "resonance".
The idea is that if something vibrates at a certain frequency it can be pushed at that frequency and the resulting motion will be large.
An example is pushing a swing. If you push it at the same frequency that it wants to swing, it moves a lot. If you push it too fast or too slow, it is more difficult to get it going.

If the engine is vibrating within a certain RPM range, it could be due to several potential causes. One common cause is an issue with engine mounts. Engine mounts are designed to hold the engine securely in place and dampen vibrations. Over time, these mounts can wear out or become damaged, leading to excessive engine movement and vibration. To check if the engine mounts are the likely cause, you can visually inspect them for any signs of damage or wear, such as cracks or leaking fluid. You can also try to gently shake the engine while someone else observes if there is excessive movement.

Another potential cause of engine vibration within a specific RPM range is an imbalance in the engine's rotating components, such as the crankshaft or flywheel. Any uneven distribution of weight in these components can cause the engine to vibrate excessively at specific RPMs. To diagnose this issue, you may need to take your vehicle to a professional technician who can use specialized tools, such as a dynamic balancer, to assess the balance of these components.

Additionally, ignition system problems can also contribute to engine vibrations. Faulty spark plugs, ignition coils, or ignition timing can cause uneven combustion and result in engine shake. To identify ignition system issues, you can inspect the spark plugs for signs of wear or damage, check the ignition coils for proper functioning, and ensure that the ignition timing aligns with the manufacturer's specifications.

Overall, if your engine is vibrating within a specific RPM range, it is recommended to consult a qualified mechanic or technician who can perform a thorough inspection and diagnose the exact cause of the vibration. They will have the necessary expertise and tools to help you pinpoint and resolve the issue effectively.