This is a phenomena that sounds unbelievable at best, but it does exist and is a common cause of many clocks stopping. Sympathetic vibration can be described as the transfer of energy from one object moving with a steady frequency to another object (initially not moving) connected to it. The objects will transfer energy very efficently if they are the same length; however, they do not have to be exactly the same length to transfer energy . A classic example is a large grandfather clock with a heavy pendulum and a case that is set up on a rug. When the weights travel downward and get even with the pendulum bob, the weights will absorb enough energy from the pendulum to either stop the clock and or start the weights swinging and stop the clock. Also the clock case can absorb energy from the pendulum because the case is not solid on the rug. Even the slightest instability can cause the clock to stop. On old cases ; if they are loose, if the case is not solid, the pend moving will cause the case to oscillate (this may or may not be visible) and the case will absorb enough energy to stop the clock. correct this by securing the case to the wall. be sure the case is solid.
If there are several clocks running on a shelf with similar pendulum lengths , and the shelf appears solid it still may be possible that the clocks on this shelf will transfer energy back and forth. The result may not be just stoppage. One or both of the clocks may not keep time or may be un-regulatable because they will affect each other.
Sympathetic vibration may be demonstrated by a little device with weights and strings. Suspend 2 weights ( at least 8 ounces ; fishing weights would work nicely ) from a small frame like a scale model of a swing set with strings. Make the strings the same length. Start one weight swinging and very soon the other one will be swinging, and the first one will be stopped. This is sympathic viabration. The less solid your frame the quicker the energy will bea transferred. The frequency with which the energy is transfered back and forth ( the weights will start and stop alternately) is related to the length of the strings and the amount of friction present and the time it takes to transfer the energy.