Vibration confinement in a general beam structure during harmonic excitations

In the present paper the Greenʹs function is utilized to provide a simple, exact and direct analytical method for analysis of a beam structure of generic boundary conditions with attached springs and/or finite masses. The aim is to confine the vibration in a certain part of the structure. The beam equation is based on the Timoshenko beam theory with corrections for shear deformation and rotary inertia effects. In the analysis the beam is driven by a harmonic external excitation. The attached springs are modeled as simple reactions that provide transverse forces to the beam, while each added mass provides a transverse force in addition to a moment at its location. These forces (moments) act as secondary forces (moments) that reduce the response caused by the external force. Numerical simulations are conducted to find the optimal masses and/or springs that confine the vibration in a certain chosen region. The results were compared to that obtained from Euler–Bernoulli beam theory. Beams that are excited by a bi-harmonic force as well as dual excitation forces are analyzed. In addition, the case when the beams are excited near resonances is discussed. Also, a method is proposed to impose a node at any desired location along the structure.