Exploring vibration isolation of a superelastic shape memory alloy cantilever beam

Abstract

Vibration of a superelastic Shape Memory Alloy (SMA)cantilever beam has been analyzed comprehensively considering end-shortening effect, geometric non-linearity and effect of material non-linearity (that is, the beam material has a non-linear stress-strain relation and Hook’s law cannot be applied)for different forcing frequency and damping ratio. Difference between vibration for two equivalent cross-sections (circular and equivalent rectangular) has also been analyzed. Soundness of the developed mathematical scheme and computer code has been amply demonstrated in the present study.

The Euler-Bernoulli’s geometrically non-linear governing equation and equation of classical theory for beam are solved numerically to generate the load-deflection relations. From these load-deflection relations, the restoring forces are chosen as best fit non-linear curves. Then this best fit non-linear curves of restoring forces are inserted into the equation of vibration that converts the equation into non-linear form which is solved numerically by the Runge-Kutta (R-K) 4th order method of integration since exact solution is not possible. Material non-linearity issue is handled in terms of a mathematical model that has been developed to solve pure bending of a beam. It should be noted here that, SMA inherently has highly non-linear stress-strain curves in tension and compression. Thus, bending moment-curvature and reduced modulus-curvature relations are obtained for a superelastic SMA circular and equivalent rectangular beams. Results are used to predict effect of material non-linearity on the response of vibration of the beam.

Some interesting results can be observed from this thesis that the amplitude of the undamped vibration considering end-shortening effect is not going to be infinite over time at resonance(β = ω/ω_n =1) which is predicted by classical theory. It is also found that increased damping greatly reduces the vibration amplitude for both the classical theory with and without end-shortening effect.

As material non-linearity is taken into account some portion of the beam is found to experience stresses (different magnitude in tension and compression) beyond proportional limit. Effect of material non-linearity becomes prominent as vibration amplitude starts to become large. Moreover, the maximum deflection of the beam is found to be much larger when material non-linearity is considered in comparison to the case of linearly elastic beam material.

The amplitude of the vibration considering material non-linearity and geometric non-linearity with end-shortening effect is largely lower than that predicted by classical theory when the damping is low. Increased damping greatly reduces the vibration.

The amplitude of the undamped vibration for circular beam is slightly higher than that of equivalent rectangular beam at resonance (β = 1) considering material non-linearity with end-shortening effect. When the geometric non-linearity issue is incorporated to the material non-linearity with end-shortening effect the discrepancy between the vibrations is reduced.

The isolation of the vibration begins when the speed ratio (β) exceeds 1.414 for classical theory with any amount of damping but it requires higher speed ratio to start the vibration isolation when end-shortening effect is included. A very interesting result is found that the speed ratio (β) varies with the damping ratio to isolate the vibration considering material non-linearity and geometric non-linearity with end-shortening effect. A higher damping ratio requires a smaller speed ratio to begin the isolation.

Key words: Cantilever Beam, End-Shortening, Equivalent Cross-section, Vibration Isolation, Transmissibility.