I don't see any obvious structural problem with your existing situation. Your proposal would certainly make for a very ridgid floor system. Just for the heck of it, maybe you should isolate the washer from the floor so it doesn't transmit this vibration (I assume the floors don't bounce). An inexpensive trick I have used to silence a washer and dryer is to cut dense rubber mats and set them under the machines. Believe it or not I use stall mats (for horses) 5/8" thick. This totally isolates the machines from the floor, and is tough enough to stand up to the weight and vibration. PS: a lot less work!