Structural Vibrations: Normal Modes in a Hand-Bell
The vibrational modes contributing most significantly to the radiated sound of hand-bells, are the bending waves travelling along the bell surface with periodic boundary conditions in the angular direction, nearly clamped boundaries near the crown, and free boundary conditions at the mouth of the bell. They are usually identified by two indices (m,n), where m counts the number of nodal lines crossing the crown, and n the number of circumferential nodal lines. The (m,0) modes, for values of m higher than a critical value, are missing. They are replaced by a mode for which the first circumferential nodal line lies very close to the mouth of the bell. These modes are designated as (m,1#) modes. Using Finite Element Analysis (FEA) a hand-bell is modeled with increasing complexity to show that both positive and negative curvatures are required in the bell wall to account for the presence of the (m,1#) modes.