A Comparative Study of Vibration Analysis of Piezoelectrically-Actuated Cantilever Beam Systems under Different Modeling Frameworks

In this research a comprehensive modeling framework for a piezoelectrically-actuated cantilever beam is developed and a detailed model and vibration analyses is performed. To achieve this goal, the governing dynamics for the system as well as boundary conditions are derived using the extended Hamilton's principle. The equations of motion of cantilever beam are derived according to the Euler-Bernoulli, Rayleigh and Timoshenko theories separately. The Euler-Bernoulli theory neglects the effects of rotary inertia and shear deformation and is only applicable to analysis of thin beams. The Rayleigh theory considers the effect of rotary inertia, while the Timoshenko theory considers the effects of both rotary inertia and shear deformation for thick beams. It is evident from the nature of discontinuous geometry of system, equation of stress-strain relationship are modified as shown in theory subsection meanwhile the natural surface in the composite (beam-piezoelectric layer) portion of the cantilever beam must be considered in this stage of calculations. Then the first five natural frequencies of this composite system are obtained by those three different theories and the results are compared. Relevant mode shapes are also drawn and effects of including rotary inertia and shear deformation are discussed for slender and stocky beams. Then, the forced vibration problem is solved and the cantilever tip deflection is obtained in which applied voltage to the piezoelectric layer is considered to be a unit-step input. The results are compared again for slender and stocky beams.