PA28 flight instructor on training flight with student reported engine malfunction inflight.

Narrative

Student and I had swapped around aircraft but had finally gotten Aircraft X pre-flighted and ready to conduct a standard private pilot flight training lesson. We picked up the current ATIS and began taxi to Runway XX via ZZZ. During taxi we noticed the aircraft was bouncing oddly but I attributed it to a bad damper or struts. We conducted a run up in the run up area in accordance with the checklist and notated no anomalies. We commenced take-off around XA:15; takeoff run was stable with no abnormal engine performance or events during takeoff roll. After rotation we turned to heading 310 towards the rail depot as prescribed in our local departure procedure. We leveled out at 2000 ft. momentarily due to traffic advisory from ATC and after traffic was no factor; we continued to climb to 2500 ft. The ceiling was at 3000 ft. in this area so I advised my student to maintain 2500 ft. for the time being. They trimmed the aircraft to maintain level flight and brought the power back to 2300 RPM. During this time I was discussing the weather with them. I told them I had some concerns with how the weather was looking. I advised them that the visible cloud tops we were able to see were much too high for us to climb over based on my observation; so I advised them that we would have to forgo stalls for this lesson and instead would do steep turns and ground reference maneuvers' and head back into ZZZ1 airport. I advised the student that I saw an area just ahead which appeared to have a higher ceiling; so I told them; once we get in that area; we would commence a climb to 2700 ft; do one or two steep turns then go back down to lower altitude to do ground ref. Once we got to this area; which was just past ZZZ2 airport; I saw some fields in the distance and told them to point the nose towards the field and commence a climb from 2500 ft. to 2700 ft. The student acknowledged and began to advance the throttles to climb. They hadn't fully advanced the throttles to full power when I suddenly heard a massive bang; almost as though something exploded. The aircraft then began shaking violently and a warning alert was going off. I was momentarily stunned by the bang; and for a second I was trying to understand what possibly went wrong. I thought perhaps I had blown a piston; however regardless of what it was; I was very concerned that vibrations would dislodge the engine out of the cowling if not break the airplane up; so I immediately retarded the throttles to see if that would stop the vibrations. The vibration damped a little but the aircraft was still shaking a lot. At this time; I had already selected a field; and let my student know we were putting it down in that field. I commenced a left turn towards the field and had a lot of difficulty with returning the aircraft level (to the right). Because the controls were so hard; I was calling out instructions to my student. I instructed them to add more trim. This time I was able to put in XXX.X and make a quick mayday call. By the time I finished my mayday call my landing field was dead ahead and I was certain I would make it; verified my speed then began calling for my flaps; which my student deployed for me. We touched down in a field and while it was bouncy due to the rows of seedlings; I was satisfied that I had full command of the steering and that we had enough field to bring the aircraft to a complete stop without the use of brakes. Once we stopped. I shut down everything; and then we exited the aircraft and contacted law enforcement for assistance.