Quadratic Potential Fields Research Articles

Chaplygin sleigh in the quadratic potential field

We study numerically the dynamics of Chaplygin sleigh under the action of the quadratic potential field. In contrast with the free Chaplygin sleigh our mechanical model manifests a complex behaviour: conservative-like chaotic regimes at low energies, coexistent pairs of chaotic attractors and repellers, mapping to each other by time-reversal symmetry, and the recently discovered phenomenon of attractor and repeller intersection, at high energies. We demonstrate that the development of attractors and repellers corresponds to a period doubling scenario, followed by their collision and instant increase in size.

Reversible hysteresis for broadband magnetopiezoelastic energy harvesting

We model and experimentally validate a nonlinear energy harvester capable of bidirectional hysteresis. In particular, both hardening and softening response within the quadratic potential field of a power generating piezoelectric beam (with a permanent magnet end mass) is invoked by tuning nonlinear magnetic interactions. Not only is this technique shown to increase the bandwidth of the device but experimental results additionally verify the capability to outperform linear resonance. Engaging this nonlinear phenomenon is ideally suited to efficiently harvest energy from ambient excitations with slowly varying frequencies.

Superposition Methods for Distributed Manipulation Using Quadratic Potential Force Fields

Planar quadratic potential fields are useful for distributed manipulation because they are readily analyzable and naturally produce predictable equilibria. This generally simplifies implementation, since feedback and control may not be necessary. Traditionally, to dynamically produce moving fields for complex manipulation tasks, these fields had to be realized by highly capable, but redundant, actuator arrays. This paper suggests a new method: using simpler devices to generate basic component fields, and superposing these fields to produce desired fields with similar degrees of freedom. This approach is particularly useful for naturally produced force fields which do not allow the dynamic moving and changing of a field, but allow for superposition, and thus, can be spatially combined to produce the desired net behaviors. A vector representation of these fields is developed and applied to two problems: how to place the fields in space to span the maximum possible configuration space; and how to generate an optimal solution to generate a desired field by a superposition of a fixed field arrangement with minimal effort from each field. We experimentally validated these methods using airflow fields based on phenomenological and time superposition. Finally, we use superposition to simplify trajectory following, by blending proportions of two fields over time and by superimposing a set of component fields, each responsible for compensating a particular dynamic term

Generation of quadratic potential force fields from flow fields for distributed manipulation

Distributed manipulation systems induce motions on objects through the application of many external forces. Many of these systems are abstracted as planar programmable force fields. Quadratic potential fields form a class of such fields that lend themselves to analytical study and exhibit useful stability properties. This paper introduces a new methodology to build quadratic potential fields with simple devices using the naturally existing phenomena of airflow, which is an improvement to the traditional use of the complicated programmable actuator arrays. It also provides a basis for the exploitation, in distributed manipulation, of natural phenomena like airflow, which require rigorous analysis and display stability difficulties. A demonstration and verification of the theoretical results for the special case of the elliptic field with airflows is also presented.

High-order field electrophoresis theory for a nonuniformly charged sphere

An electrophoresis theory is developed for a rigid sphere in a general nonuniform electric field. The zeta potential distribution and the double-layer thickness are both arbitrary. The zeta potential of the sphere is assumed to be small so that the deformation of the double layer can be neglected. Explicit expressions for the translational and rotational velocities of the sphere are derived in terms of the multipole moments of the zeta potential distribution and the tensor coefficients of the applied electric field. The presence of the kth-order component in the electrical potential field applied to the sphere results in a translation of the sphere only when the sphere possesses the ( k−1)th- or ( k+1)th-order multipole moments of the zeta potential distribution. In addition, the kth-order component in the electrical potential field causes a rotation of the sphere only when the sphere possesses the kth-order moment of the zeta potential distribution. As an illustrative example for the utility of our theory, we theoretically devise an electrophoresis analysis scheme for estimating the dipole moment of a dipolar sphere by observing the electrophoretic translation of the sphere in a quadratic potential field.

On the dynamics of the four-dimensional rigid body in a quadratic potential field

We study the nondegenerated solutions of the rotation of a four-dimensional rigid body in a quadratic potential field. This problem has 6 degrees of freedom. We obtain 143 topologically different solutions and explicit formulas in Prym theta-functions.

The combined use of SCF-MO calculations and experimental data in the evaluation of quadratic force fields: MOCIC potential functions for fluoroform, methyl acetylene, and methyl cyanide

A semiempirical method combining SCF-MO calculations and limited vibrational data has been employed to evaluate the completely general quadratic potential fields of fluoroform, methyl acetylene, and acetonitrile. MOCIC (molecular orbital constraint using interaction coordinates) potential fields are presented for gas phase molecules of intermediate size. Here general harmonic force fields or excellent approximations utilizing extensive experimental data are available as standards. A statistical evaluation of the interaction potentials shows that there is some improvement in going from MNDO or ab initio SCF-MO force fields to the MOCIC functions which reliably reproduce the off-diagonal vibrational potential constants in most instances. The MOCIC primary compliants are excellent approximations of their vibrational counterparts, as expected. Comparison of the calculated isotopic frequencies, Coriolis coupling constants, and centrifugal distortion constants for the SCF-MO, MOCIC, and vibrational spectroscopic potential fields with the corresponding experimental values also shows MOCICs reliability for molecules with many interaction potentials. There is substantial improvement in the calculated isotopic frequency shifts and centrifugal distortion constants in going from SCF-MO to MOCIC functions.

Gaussian-Wave-Packet Dynamics in Uniform Magnetic and Quadratic Potential Fields

For the case of uniform magnetic field and a potential quadratic in its coordinates an electron state originally represented by a Gaussian wave packet remains Gaussian in its subsequent evolution. Under these conditions the state is completely characterized by the first-and second-order quantum means, that is, quantities such as $〈{q}_{i}〉$, $〈{p}_{i}〉$ and $〈{q}_{i}{p}_{j}〉$, $〈{p}_{i}{p}_{j}〉$, respectively. A tensor-product relationship is shown to exist between the problem of the determination of the second- and first-order means which facilitates the solution. Two problems are treated in detail: (i) an isotropic wave packet in an isotropic potential well and (ii) the dynamics of a wave packet on a quadratic saddle point. In the former case, the wave packet "spins" and its radius pulsates as its mean follows the classical trajectory. In the second the magnetic field effect on tunneling through such barriers is studied and found to be small at attainable field strengths unless the effective electronic mass is very small. A method for the numerical solution for the time-dependent Schr\"odinger equation, termed the particle method and based on the hydrodynamic analogy to that equation, is here extended to include magnetic fields. It is applied to the two problems just described and found to agree well with the analytical solutions.