Francis C. Moon

Modal sensors/actuators

A piezoelectric laminate theory that uses the piezoelectric phenomenon to effect distributed control and sensing of structural vibration of a flexible plate has been used to develop a class of distributed sensor/actuators, that of modal sensors/actuators. The one-dimensional modal sensors/actuator equations are first derived theoretically and then examined experimentally. These modal equations indicate that distributed piezoelectric sensors/actuators can be adopted to measure/excite specific modes of one-dimensional plates and beams. If constructed correctly, actuator/observer spillover will not be present in systems adopting these types of sensors/actuators. A mode 1 and a mode 2 sensor for a one-dimensional cantilever plate were constructed and tested to examine the applicability of the modal sensors/actuators. A modal coordinate analyzer which allows us to measure any specific modal coordinate on-line real-time is proposed. Finally, a way to create a special two-dimensional modal sensor is presented.

A magnetoelastic strange attractor

Abstract Experimental evidence is presented for chaotic type non-periodic motions of a deterministic magnetoelastic oscillator. These motions are analogous to solutions in non-linear dynamic systems possessing what have been called “strange attractors”. In the experiments described below a ferromagnetic beam buckled between two magnets undergoes forced oscillations. Although the applied force is sinusoidal, nevertheless bounded, non-periodic, apparently chaotic motions result due to jumps between two or three stable equilibrium positions. A frequency analysis of the motion shows a broad spectrum of frequencies below the driving frequency. Also the distribution of zero crossing times shows a broad spectrum of times greater than the forcing period. The driving amplitude and frequency parameters required for these non-periodic motions are determined experimentally. A continuum model based on linear elastic and non-linear magnetic forces is developed and it is shown that this can be reduced to a single degree of freedom oscillator which exhibits chaotic solutions very similar to those observed experimentally. Thus, both experimental and theoretical evidence for the existence of a strange attractor in a deterministic dynamical system is presented.

Capacitance Based Tunable Micromechanical Resonators

We present actuators which tune the resonant frequency of micromechanical oscillators. Experimental results show resonant oscillations from 7.7% to 146% of the original resonant frequency. Numerical results substantiate these results. Two failure modes have been identified which limit

Chaotic vibrations of a beam with non-linear boundary conditions

Abstract Forced vibrations of an elastic beam with non-linear boundary conditions are shown to exhibit chaotic behavior of the strange attractor type for a sinusoidal input force. The beam is clamped at one end, and the other end is pinned for the tip displacement less than some fixed value and is free for displacements greater than this value. The stiffness of the beam has the properties of a bi-linear spring. The results may be typical of a class of mechanical oscillators with play or amplitude constraining stops. Subharmonic oscillations are found to be characteristic of these types of motions. For certain values of forcing frequency and amplitude the periodic motion becomes unstable and nonperiodic bounded vibrations result. These chaotic motions have a narrow band spectrum of frequency components near the subharmonic frequencies. Digital simulation of a single mode mathematical model of the beam using a Runge-Kutta algorithm is shown to give results qualitatively similar to experimental observations.

Subcritical Hopf Bifurcation in the Delay Equation Model for Machine Tool Vibrations

We show the existence of a subcritical Hopf bifurcation in thedelay-differential equation model of the so-called regenerative machine toolvibration. The calculation is based on the reduction of the infinite-dimensional problem to a two-dimensional center manifold. Due to the specialalgebraic structure of the delayed terms in the nonlinear part of the equation,the computation results in simple analytical formulas. Numerical simulationsgave excellent agreement with the results.

High‐speed rotation of magnets on high Tc superconducting bearings

Levitation and rotation of cylindrical rare‐earth magnets on yttrium‐barium‐copper‐oxide superconducting bearings has been sustained at speeds of up to 120 000 rpm with a surface speed of 40 m/s. The decay of the free rotation rate has been measured at both atmospheric pressure and a partial vacuum to 2.6 μm Hg. The decay measurements in vacuum indicate that the flux drag torques are constant and independent of speed. The magnetic shear stress on the rotor magnet is estimated to be 150 dyn/cm2. It is believed that drag torques on rotating magnets are related to asymmetry in the flux density pattern of the magnet.

Laminated piezopolymer plates for torsion and bending sensors and actuators

A set of piezopolymer devices has been developed based on a composite laminate theory for piezoelectric polymer materials. By using different combinations of ply angles and electrode patterns, a piezopolymer/metal shim plate structure was built that exhibited both bending and torsion deformation under an applied electric field. A set of torsion‐beam sensor structures was also built that could distinguish between bending and torsion or between different vibration modes. These devices were based on a general theory of piezoelectric laminates. The experimental results agreed quite closely with the theoretical predictions. These integrated sensor–actuator devices may find application in the control of microactuators or may be used for modal control of larger continuous structures.

Hysteretic levitation forces in superconducting ceramics

Magnetization forces between small permanent magnets and bulk high Tc superconducting ceramic materials have been measured and show marked hysteretic behavior as a function of the distance between the magnet and superconductor. Magnetic forces of up to 2500 dyn, both normal and tangential to the superconductor surface, have been made. Near‐reversible forces are obtained for small motions of the magnet, however, which are believed to be related to flux pinning.

Strange Attractors and Chaos in Nonlinear Mechanics

We review several examples of nonlinear mechanical and electrical systems and related mathematical models that display chaotic dynamics or strange attractors. Some simple mathematical models — iterated piecewise linear mappings — are introduced to explain and illustrate the concepts of sensitive dependence on initial conditions and chaos. In particular, we describe the role of homoclinic orbits and the horseshoe map in the generation of chaos, and indicate how the existence of such features can be detected in specific nonlinear differential equations.

Dynamic magnetic forces in superconducting ceramics

Dynamic forces between small permanent magnets and bulk high‐Tc superconducting ceramic materials have been measured, including magnetic stiffness and damping effects. The vibration frequencies and the related magnetic stiffness show a large dependence on the magnet‐superconductor distance. The dynamic magnetic stiffness is shown to be related to small static reversible magnetic forces, but is not correlated with the static hysteretic forces. The magnetic damping is inversely related to the magnet‐superconductor distance and can critically damp the oscillations at small gaps. These data also show that the flux pinning forces depend on the prior flux history in the superconductor.