Parameter Sensitivity Analysis of Piezoelectrically-Actuated Flexural/Torsional Vibrating Beams

A numerical parameter sensitivity analysis is performed on the bending and torsional vibrations of a flexural-torsional vibrating beam gyroscope model. The gyroscope analyzed in this work is comprised of a rotating cantilever beam with a point-mass attached to its free end and a piezoelectric actuator fixed to a portion of its length. The governing equations of motion are derived using extended Hamilton’s principle and the steady-state magnitude response of the system is obtained through frequency domain methods. A sensitivity analysis is then carried out for the parameters including rotational speed of the base, the length of the beam, the location of the piezoelectric patch, and the location of the added point mass along the beam’s length. It is observed that, in the region surrounding specific configurations, small variations in the rotation rate, beam length and the location of the piezoelectric patch will result in significant changes to the amplitudes of the coupled vibrations and produce peaks in the sensitivity curves. Further, the amplitude of vibration tends to increase as the location of the added point-mass is moved closer to the free end. Generally, the bending modes are more sensitive to all of these parameter variations than are the torsional modes.

The Exact Frequency Equations for the Euler-Bernoulli Beam Subject to Boundary Damping

The Euler-Bernoulli (E-B) beam is the most commonly utilized model in the study of vibrating beams. The exact frequency equations for this problem, subject to energy-conserving boundary conditions, are well-known; however, the corresponding dissipative problem has been solved only approximately, via asymptotic methods. These methods, of course, are not accurate when looking at the low end of the spectrum. Here, we solve for the exact frequency equations for the E-B beam subject to boundary damping. Numerous numerical examples are provided, showing plots of both the complex wave numbers and the exponential damping rates for the first five frequencies in each case. Some of these results are surprising.

The Exact Frequency Equations for the Rayleigh and Shear Beams with Boundary Damping

While the Euler-Bernoulli beam is the most commonly utilized model in studying vibrating beams, one often requires a model that captures the additional effects of rotary inertia or deformation due to shear. The Rayleigh beam improves upon the Euler-Bernoulli by including the former effect, while the shear beam is an improvement that includes the latter. While all of these problems have been well studied when subject to energy-conserving boundary conditions, none have been solved for the case of boundary damping. We compute the exact frequency equations for the Rayleigh and shear beams, subject to boundary damping and, in the process, we find interesting connections between the two models, despite their being very different.