Stable Particle Motion Research Articles

Simulating quasi-integrable optics with space charge in the IBEX Paul trap

The intensity frontier has called for new initiatives in hadron accelerator design in order to accommodate space charge dominated beams. Octupoles are often used to damp beam instabilities caused by space charge, however the insertion of octupole magnets leads to a nonintegrable lattice which reduces the area of stable particle motion. One proposed solution is Quasi-Integrable optics (QIO), where the octupoles are inserted between sections of a specific lattice insertion called a T-insert. An octupole with a strength that scales as 1/β 3(s) is applied in the drift region, where the horizontal and vertical beta functions are equal, to create a time independent octupole field. This leads to a lattice with a time-independent Hamiltonian which is robust to small perturbations. IBEX is a Paul trap which allows the transverse dynamics of a collection of trapped particles to be studied, mimicking the propagation through multiple quadrupole lattice periods, whilst remaining stationary in the laboratory frame. In order to test QIO at the IBEX experiment, it has recently undergone an upgrade to allow for the creation of octupole fields. We present our design of the IBEX experiment upgrade along with simulation results of our proposed experiment to test QIO with space charge.

Smoothed Particle Hydrodynamics Simulations of Dam-Break Flows Around Movable Structures

In this paper, 3D weakly compressible smoothed particle hydrodynamics (WCSPH) and incompressible smoothed particle hydrodynamics (ISPH) models are used to study dam-break flows impacting either a fixed or a movable structure. First, the two models’ performances are compared in terms of CPU time efficiency and numerical accuracy, as well as water surface shapes and pressure fields. Then, they are applied to investigate dam-break flow interactions with structures placed in the path of the flood. This study found that the ISPH modeling approach is slightly superior to the WCSPH approach because more stable particle motion and pressure distribution can be achieved with a reasonable CPU load.

Stochastic Resonance with a Joint Woods-Saxon and Gaussian Potential for Bearing Fault Diagnosis

This work aims for a new stochastic resonance (SR) model which performs well in bearing fault diagnosis. Different from the traditional bistable SR system, we realize the SR based on the joint of Woods-Saxon potential (WSP) and Gaussian potential (GP) instead of a reflection-symmetric quartic potential. With this potential model, all the parameters in the Woods-Saxon and Gaussian SR (WSGSR) system are not coupled when compared to the traditional one, so the output signal-to-noise ratio (SNR) can be optimized much more easily by tuning the system parameters. Besides, a smoother potential bottom and steeper potential wall lead to a stable particle motion within each potential well and avoid the unexpected noise. Different from the SR with only WSP which is a monostable system, we improve it into a bistable one as a general form offering a higher SNR and a wider bandwidth. Finally, the proposed model is verified to be outstanding in weak signal detection for bearing fault diagnosis and the strategy offers us a more effective and feasible diagnosis conclusion.

Stochastic resonance with Woods–Saxon potential for rolling element bearing fault diagnosis

This paper proposes a weak signal detection strategy for rolling element bearing fault diagnosis by investigating a new mechanism to realize stochastic resonance (SR) based on the Woods–Saxon (WS) potential. The WS potential has the distinct structure with smooth potential bottom and steep potential wall, which guarantees a stable particle motion within the potential and avoids the unexpected noises for the SR system. In the Woods–Saxon SR (WSSR) model, the output signal-to-noise ratio (SNR) can be optimized just by tuning the WS potential's parameters, which delivers the most significant merit that the limitation of small parameter requirement of the classical bistable SR can be overcome, and thus a wide range of driving frequencies can be detected via the SR model. Furthermore, the proposed WSSR model is also insensitive to the noise, and can detect the weak signals with different noise levels. Additionally, the WS potential can be designed accurately due to its parameter independence, which implies that the proposed method can be matched to different input signals adaptively. With these properties, the proposed weak signal detection strategy is indicated to be beneficial to rolling element bearing fault diagnosis. Both the simulated and the practical bearing fault signals verify the effectiveness and efficiency of the proposed WSSR method in comparison with the traditional bistable SR method.

Stable Particle Motion in a Linear Accelerator with Solenoid Focusing

We derived the equation governing stable particle motion in a linear ion accelerator containing discrete rf and either discrete or continuous solenoid focusing. We found for discrete solenoid focusing that cos ? = (1 + d?) cos ?/2 + (l?/? - d?/21 - ??d2/41) sin ?/2 ? = 1/f and l + 2d = s?, where ?, ?, f, l, and d are the phase advance per cell, precession angle in the solenoid, focal length of the rf lens, length of the solenoid in one cell, and the drift distance between the center of the rf gap and the effective edge of the solenoid. The relation for a continuous solenoid is found by setting d equal to zero. The boundaries of the stability region for ? vs ? with fixed l and d are obtained when cos ? = +1.

Note on the magnetic-moment adiabatic invariant for particle motion in a dipole magnetic field

The magnetic-moment adiabatic-invariant series is evaluated through second order in the expansion parameter ϵ = ρ/L, i.e. the ratio of the cyclotron radius ρ = pc/eB to the McIlwain shell parameter, for the particular case of a particle at the equatorial plane of a dipole magnetic field. The resulting expression for the magnetic-moment series is compared with computed values of the magnetic moment for particle orbits in a dipole magnetic field obtained by numerically integrating the equations of motion. Orbits have been calculated for particles with 1.22 × 10−2 < ϵ < 8.29 × 10−2. Under conditions of stable particle motion, the numerical results verify the constancy of the magnetic-moment series through O(ϵ²) to accuracies σ (SD) = 0.026–3.2% depending on the value of ϵ.

The behaviour of particles in a sinusoidal velocity field

A non-linear Langevin equation has been developed that represents the general behaviour of free particles in a sinusoidal velocity field. Analysis shows that a linear drag law, in which the frictional force is directly proportional to the relative particle-fluid velocity (Stokes’s law), does not result in a stable particle motion. However, a non-linear relation in which the drag is proportional to the square of the relative velocity (Newton’s law) leads to the Mathieu equation, wiien it may be shown that stable particle trajectories may occur in certain ranges of frequency and amplitude. Stability criteria based on the Mathieu equation are applied to bubble-liquid systems in the sonic and ultrasonic regions, to liquid drop systems in pulse columns and to solid-liquid systems by using drag coefficients obtained from measurements of terminal velocities. The agreement between the theoretical and observed frequencies indicates that the velocity profile around a particle in a sinusoidal field is approximated by a quasi-steady state distribution and that transient effects may be neglected for the separated flow conditions considered in the present communication.