Modeling and Analysis of a Cantilever Beam Tip Mass System

We model the nonlinear dynamics of a cantilever beam with tip mass system subjected to different excitation and exploit the nonlinear behavior to perform sensitivity analysis and propose a parameter identification scheme for nonlinear piezoelectric coefficients.

First, the distributed parameter governing equations taking into consideration the nonlinear boundary conditions of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. We solve the distributed parameter and discretized equations separately using the method of multiple scales. We determine that the cantilever beam tip mass system subjected to parametric excitation is highly sensitive to the detuning. Finally, we show that assuming linearized boundary conditions yields the wrong type of bifurcation.

Noting the highly sensitive nature of a cantilever beam with tip mass system subjected to parametric excitation to detuning, we perform sensitivity of the response to small variations in elasticity (stiffness), and the tip mass. The governing equation of the first mode is derived, and the method of multiple scales is used to determine the approximate solution based on the order of the expected variations. We demonstrate that the system can be designed so that small variations in either stiffness or tip mass can alter the type of bifurcation. Notably, we show that the response of a system designed for a supercritical bifurcation can change to yield a subcritical bifurcation with small variations in the parameters. Although such a trend is usually undesired, we argue that it can be used to detect small variations induced by fatigue or small mass depositions in sensing applications.

Finally, we consider a cantilever beam with tip mass and piezoelectric layer and propose a parameter identification scheme that exploits the vibration response to estimate the nonlinear piezoelectric coefficients. We develop the governing equations of a cantilever beam with tip mass and piezoelectric layer by considering an enthalpy that accounts for quadratic and cubic material nonlinearities. We then use the method of multiple scales to determine the approximate solution of the response to direct excitation. We show that approximate solution and amplitude and phase modulation equations obtained from the method of multiple scales analysis can be matched with numerical simulation of the response to estimate the nonlinear piezoelectric coefficients.