Axial Oscillations Research Articles

Influence of Heat Treatment of Nickel-Titanium Instruments on Cyclic Fatigue Resistance in Simulated Curved Canals.

The aim of this study was to compare the cyclic fatigue of heat-treated and non-treated instruments. Twenty instruments with and without heat treatment from Bassi Logic were evaluated (n = 10). All instruments were subjected to dynamic cyclic fatigue through continuous rotations inside a stainless-steel tube (1.4-mm diameter, 9-mm curvature with 6-mm radius, and 90° angle) using a custom-made device, which performed 1 axial oscillation every 2 seconds with an amplitude of 3 mm, powered by a torque-controlled motor (Silver Reciproc, VDW, Germany), with speed adjusting to 950 rpm and torque to 4 N, according to manufacturer's guidance. The groups were compared using the Mann-Whitney test. The fracture time of the heat-treated instruments (97.20 ± 39.94 second and non-treated instruments (14.30 ± 6.78 seconds had statistical differences [p < 0.05]). Heat-treated instruments were 6.8 times more resistant to fatigue than non-treated instruments (p < 0.05). Heat treatment provides increased fatigue resistance of NiTi alloy with the same design.

Temporal instability of confined three-dimensional liquid jet with heat and mass transfer under longitudinal acoustic oscillations

The temporal instability of a confined viscous liquid jet surrounded by high-speed co-flowing viscous gas phase is studied in this work. The effect of the longitudinal acoustic oscillations, which is regarded as gas axial velocity oscillations, is also considered. The heat and mass transfer is characterized by the ratio between conduction heat flux and the evaporation heat flux; then, an explicit dispersion relation equation is obtained. The results suggest that more than one unstable region appears because of the gas velocity oscillations, including Kelvin–Helmholtz (K–H) instability and parametric instability regions. Increasing the forcing frequency enhances the K–H instability, while it has a stabilizing effect on the parametric instability. In addition, the liquid jet tends to be more unstable in non-axisymmetric modes when the gas rotating strength is strong. Although the gas viscosity has a destabilizing effect on the gas–liquid interface, the destabilizing effect is weak due to the low viscosity of the gas phase. According to the linear instability theory, the dominant wavenumber will locate in the most unstable region. Moreover, the parametric instability in non-axisymmetric modes may be observable when the Weber number is large.

Molecular sensors producing circularly polarized luminescence responses

Circularly polarized luminescence (CPL) refers to spontaneous emission with the oscillation axis of the emission's electric dipole rotating either right handedly or left handedly along the propagation axis. CPL provides a unique principle in luminescence visualization of target analytes: both the direction and magnitude of light polarization are sensitive to an asymmetric environment around a luminophore, which enables chirality sensing. In addition, CPL spectra are usually less complex than electric circular dichroism spectra because they emerge only from the lowest excited state. To exploit these benefits, researchers have vigorously developed CPL probes. This minireview provides the basic principles for creating molecular CPL activity, including asymmetric exciton coupling and helical intramolecular charge transfer. Emphasis is placed on molecular design. Analyte interactions can perturb the magnitude, reverse the polarization direction, or shift the peak wavelength of CPL, thereby enabling tractable strategies for creating CPL sensors. This minireview outlines selected examples of CPL sensors for physical stimuli, including changes in external magnetic fields, solvent polarity, and temperature. Probes capable of detecting chemical species, such as protons, metal ions, anions, amino acids, nucleosides and DNA, reactive oxygen species, and humidity, are also highlighted. These examples demonstrate the unique sensing utility of CPL sensors. We hope that this minireview will stimulate future research interest toward developing advanced CPL sensors with applications in bioimaging.

Past and Future Practical Solutions for Torsional Vibration Damping in Vehicle Industry

In addition to material and production costs, consumption and emission limits, the requirements for performance, efficiency and space utilization must be met when it comes to the design of today's internal combustion engines for the automotive industry. As a result, three new design trends have been emerged (based on J. Pfleghaar and B. Lohmann's paper in 2013): 1. downsizing: reduction of engine size (number of pistons and stroke) for fuel and space-saving and CO2 emission reduction purposes, 2. downspeeding: reduction of engine speed to save fuel, which necessarily entails significantly higher torques being generated and transmitted in the engine, 3. turbo supercharging: increasing the pressure and compression ratio in the engine piston cylinder to cover the increased torque demand, which is accompanied by NOx gas emissions. Due to these new design trends, significant transverse, axial, and torsional oscillations can occur on the engine's crankshaft. To avoid power loss and fatigue due to the torsional oscillations, a torsional vibration damper is advised to be installed on the free end of the crankshaft or integrated into the flywheel. This review paper focuses on the possible reasons for torsional vibrations, the applied methods used to dampen them, and expected future trends.

Fluid and hybrid simulations of the ionization instabilities in Hall thruster

Low-frequency axial oscillations in the range of 5–50 kHz stand out as a pervasive feature observed in many types of Hall thrusters. While it is widely recognized that the ionization effects play the central role in this mode, as manifested via the large-scale oscillations of neutral and plasma density, the exact mechanism(s) of the instabilities remain unclear. To gain further insight into the physics of the breathing mode and evaluate the role of kinetic effects, a one-dimensional time-dependent full nonlinear low-frequency model describing neutral atoms, ions, and electrons is developed in full fluid formulation and compared to the hybrid model in which the ions and neutrals are kinetic. Both models are quasi-neutral and share the same electron fluid equations that include the electron diffusion, mobility across the magnetic field, and the electron energy evolution. The ionization models are also similar in both approaches. The predictions of fluid and hybrid simulations are compared for different test cases. Two main regimes are identified in both models: one with pure low-frequency behavior and the other one, where the low-frequency oscillations coexist with high-frequency oscillations in the range of 100–200 kHz, with the characteristic time scale of the ion channel fly-by time, 100–200 kHz. The other test case demonstrates the effect of a finite temperature of injected neutral atoms, which has a substantial suppression effect on the oscillation amplitude.

Novel Configuration for a C-Band Axial Vircator With High Output Power

We present a novel configuration for an axial virtual cathode oscillator (Vircator) operating in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${C}$ </tex-math></inline-formula> -band and designed for high output power applications. In order to enhance the efficiency, we have employed a geometry with up to seven reflectors along the drift space. While in previous research works this optimization process has been performed only varying the reflectors radii while using a fixed distance between reflectors, in our work we propose an optimization in terms of both reflectors radii and distances. Our optimized structure for the axial Vircator is able to provide an efficiency up to 15.6% and an output power of 1.6 GW in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${C}$ </tex-math></inline-formula> -band, with a working voltage of 510 kV and a cathodic current of 24 kA.

Analysis of supercritical heat transfer in horizontal helical tube with internal roughness

• Supercritical flow in horizontal helical tube with internal roughness is analyzed. • Circumferential non-uniformity and axial oscillation of T w are explored. • The internally ribbed roughness and concaved roughness are compared. • The effects of spiral roughness, centrifugal force, and buoyancy are revealed. Helically coiled tube with smooth wall (HSW) has been widely used in the fields of supercritical air conditioning systems and supercritical heat pump systems due to its compact structure and excellent heat transfer performance. However, its large circumferential non-uniformity and intense axial oscillation will bring to local stress concentration and instability of heat transfer which are both responsible for system accidents. To alleviate the above two crises, the spiral inner wall roughness is proposed to be introduced into HSW. Up to now, the performance of helically coiled tube with internally ribbed roughness (HRW) has been seldom reported and the effectivenesses of inner roughness with different structures on suppressing circumferential non-uniformity and axial oscillation are still unclear. In this work, the SST k - ω model is employed to simulate the turbulent flow of s-R1234ze(E) in horizontal HRW to fill these gaps. Firstly, the effectivenesses of different rib heights, rib lift angles, rib widths, rib numbers, and rib base angles on restraining circumferential non-uniformity and axial oscillation are explored. The influences of wall thickness are also contained. Subsequently, the internally ribbed roughness and concaved roughness are compared and evaluated according to the overall comprehensive performance criterion. Finally, the effects of spiral roughness, centrifugal force, and gravitational buoyancy force on the flow field distributions are also revealed.

Congregation of particles on a plane boundary due to the flow induced by an oscillating sphere

The complex flow-field associated with a spherical structure oscillating adjacent to a solid boundary is significant in ocean engineering. The dynamic interaction between the flow induced by an oscillating sphere and the boundary is evident not only in terms of flow-field deformation around the structures but also on the boundary in the form of shear stress modifications. The present study aims to understand the influence of a plane boundary on the shear stress distributions and particle motion near the boundary induced by the flow due to a submerged sphere oscillating in a viscous fluid. Experiments are conducted using a non-intrusive flow visualization technique, and three-dimensional direct numerical simulations are used to link the time-mean results of the experiments to the simulated Lagrangian particle drift on the plane boundary. The dye tracer in the experiments and the Lagrangian particles in the simulation congregate at the same location. This stagnation location coincides with a ring of zero mean shear stress centered on the point on the plane boundary through which the axis of oscillation of the sphere passes. This ring of zero mean stress is termed the congregation zone. The radius of the particle congregation zone is investigated as a function of the sphere oscillating amplitude, Reynolds number, and the distance between the center of the sphere and the plane boundary. Furthermore, a systematic power scaling law is established for the radius of the congregation zone from the control parameters.

Investigation of the Optimum Diameter of the Ring Reflector for an Axial Virtual Cathode Oscillator

The optimum hole diameter of a ring reflector was experimentally investigated using an axial virtual cathode oscillator (vircator) to enhance its microwave power. The ring reflector enhances the microwave power from the axial vircator by forming a cavity. The ring reflector was installed 9 mm behind the anode. The optimum hole diameter of the ring reflector was analyzed through simulations and experiments by changing the diameter from 60 mm to 160. PIC simulations show that the maximum peak microwave power was generated when the hole diameter was 116 mm and enhanced by 210%. The experiments show similar results to the simulations. The largest peak maximum power was 23.67 MW when the hole diameter was 120 mm. The simulations show that the dominant microwave frequency was formed between 5.33 GHz and 6.7 GHz. The experiments show that the dominant microwave frequency was formed between 5.3 GHz and 5.8 GHz. The frequency trend was approximately similar to that of the simulation results. However, the trend depending on the hole diameter was not as obvious as in the simulations. Although the frequency change was not as clear as in the simulations, experiments show that the hole diameter of the ring reflector affects the vircator operation.

Lean blowout characteristics of spray flame in a multi-swirl staged combustor under different fuel decreasing rates

Lean Blow Out (LBO) poses a significant safety hazard when occurring in aero-engines. Understanding the lower stability limits of gas turbine combustors and the characteristics of spray flame close to LBO are imperative for safe operation. The objective of this work is to evaluate the effects of fuel decreasing rates and pressure drops of the injector on LBO performances in a multi-swirl staged combustor equipped with an airblast injector. A set of hardware and control system was developed to realize a user-defined fuel supply law. High-speed imaging was applied to record complete LBO processes under the conditions of linear fuel reduction and stable airflow. Partical Image Velocimetry (PIV) and Planar Mie (PMie) scattering were used to acquire the flow fields and spray fields under non-reacting conditions. Experimental results have shown that LBO limits extend to leaner conditions as the pressure drop of the injector increases. With an increase of the fuel decreasing rate, the exhaust temperature before flame extinction increases, and the LBO Fuel-to-Air-Ratio (FAR) decreases. The time evolution of the integral CH* intensity conforms to a linear function during the LBO process. Proper Orthogonal Decomposition (POD) was used to analyze the dynamic characteristics of lean-burn flames. Under different fuel decreasing rates and pressure drops of the injector, flames close to LBO present similar modal spatial distributions, alternately appearing axial, radial, high-order axial, and high-order radial oscillations.

Effect of moving-orientation of the piezoelectric fan on thermal characteristics of the concave surfaces with different curvature

The present work aims at numerically investigating on the flow and heat transfer performances induced by vibrating piezoelectric fan for cooling heated concave surfaces. Two main relevant parameters, relative curvature (K) of concave surface and moving-orientation of the piezoelectric fan, are taken into consideration. The relative curvature (K) is varied from 0.01 to 0.25. The moving-orientations of the piezoelectric fan include transverse oscillation and axial oscillation. The influence mechanism of these parameters are analyzed in terms of the time-averaged heat transfer coefficient, instantaneous streamlines and isosurface of λ2. The heat transfer results indicate that the axial oscillation is more favorable for improving the heat transfer performance than the transverse oscillation, especially in the zone near the fan vibration envelope. Meanwhile, the flow field reveals that the relative curvature of the concave surfaces influences the flow feature under longitudinal configuration substantially.

Investigation of an Axial Virtual Cathode Oscillator with an Open-Ended Coaxial Cathode

A cathode with an open-ended coaxial structure is experimentally investigated using an axial virtual cathode oscillator (vircator). To enhance the microwave power output, an open-ended coaxial cathode is installed in the axial vircator. The proposed cathode is designed based on the reciprocating frequency of the vircator. The operation features of an axial vircator with a solid cathode, an annular cathode, and an open-ended coaxial cathode are comparatively analyzed through simulations and experiments. Three cathodes are machined using graphite. A stainless steel mesh with a transparency of 70% is used as an anode. The anode-to-cathode gap is fixed to 6 mm. The vircator is driven using a 10-stage PFN-Marx generator with a characteristic impedance of 31 Ω. The PFN-Marx generator applies -150 kV voltage pulses with a 170–200 ns pulse width into the vircator. The microwave power from the solid and annular cathodes is 11.22 MW and 11.27 MW, respectively. The proposed cathode generates a microwave with a power of 12.65 MW while enhancing the microwave power by 13% compared with the solid and annular cathodes. The proposed cathode shows a frequency shift to 3.4 GHz, which is a much lower frequency than that of the solid cathode at 6.34 GHz.

Experimental and Numerical Investigation of Recirculation Structures in Isothermal Swirling Coaxial Jet

Abstract Three-dimensional (3D) unsteady Reynolds-averaged Navier–Stokes (URANS) simulations are conducted in a coaxial swirling jet. Two distinct types of recirculation zones (RZ) relevant to coaxial swirling jets are considered based on the modified Rossby number Rom, which is known to represent the ratio of axial velocity deficit between two coaxial streams to the characteristic tangential velocity at the nozzle exit. The two flow states studied are Rom&amp;gt;1 and Rom≤1. The former is characterized by regions of high strain (especially in the shear layer between central and coaxial jet). It is found in this study that renormalization group (RNG) k−ɛ model is the suitable model for 3D URANS simulation of Rom&amp;gt;1 flow state. This is attributed to the model's ability to simulate flow regions that are heavily strained. The simulated results are compared with two-dimensional laser Doppler velocimetry measurements conducted as part of this study. For the flow states Rom≤1, which are characterized by the dominance of radial pressure gradient arising due to rotational (swirling) effect over the pressure gradient due to axial velocity deficit, the Reynolds stress model (RSM) is found best to simulate the flow topologies and mean and turbulent quantities. The time-averaged results obtained from optimized turbulent models are employed to gain insights into the fluid mixing phenomenon in these RZs. The unsteady axial velocity fluctuations obtained from both experiments and URANS simulations are analyzed in the frequency domain to gain insights into dominant axial oscillations prevalent in RZs.

Asymptotic theory of a flapping wing of a circular cross-section

The paper is devoted to the study of the propulsive motion of a flapping wing of a circular cross-section performing translational–rotational oscillations in a viscous incompressible fluid. To describe the flow past the wing, the unsteady Navier–Stokes equation is solved. Using the method of asymptotic expansions for the case of small amplitudes of oscillations, an analytical solution of the problem is constructed in the first two terms. It is shown that the nonlinear interaction of time harmonics of translational and rotational oscillations causes secondary flows (steady streaming) that make the wing move in the direction perpendicular to the axis of translational oscillations. For the case of cruising motion, when the average hydrodynamic force acting on the wing is equal to zero, the dependence of the average speed on the dimensionless oscillation parameters is found. The results show that for relatively large angles of rotation the cruising speed of a flapping wing can be as high as the velocity amplitude of translational oscillations. The limits of applicability of the asymptotic theory are investigated using direct numerical simulations. Numerical data demonstrate that the theory well describes the flow past the wing in a wide range of dimensionless amplitudes, frequencies and angles of rotation. In conclusion, the efficiency of the propulsion system is evaluated. It is shown that in terms of relative energy consumption, a cylindrical flapping wing can be attributed to the most efficient propulsors in the range of Reynolds numbers $Re \sim 10^2\text {--}10^3$ .

Experimental investigations on coherent flow structures in acoustically excited swirling flames using temporally-separated dual-plane Stereo-PIV

A temporally-separated dual-plane stereoscopic particle image velocimetry (PIV) technique is utilized to investigate the dynamics of periodically-excited vortices in a swirling flame under four experimental conditions. The time-averaged results under the unexcited condition are analyzed to verify the accuracy of the method. For the excited conditions, Q criteria is used to extract and classify the vortex structures within the swirling flame. It is found that under different excitation frequencies of the same amplitude, the outer vortex rings (OVRs) and the inner vortex rings (IVRs) show opposite rotational characteristics. While the excitation frequency has little influence on the central vortices induced by the swirler, the amplitude does have a great influence on the development trajectory of the central vortices. Furthermore, the amplitude and phase of oscillation velocities in three directions are successfully decomposed by using sine curve fittings at each point. A periodic acceleration and deceleration process is observed only in the axial direction. Besides the acoustic wave, it is found that the vortex rings (both OVRs and IVRs) also contribute to the axial velocity oscillations.

Axially-deformed solution of the Skyrme-Hartree-Fock-Bogoliubov equations using the transformed harmonic oscillator basis (IV) hfbtho (v4.0): A new version of the program

We describe the new version 4.0 of the code hfbtho that solves the nuclear Hartree-Fock-Bogoliubov problem by using the deformed harmonic oscillator basis in cylindrical coordinates. In the new version, we have implemented the restoration of rotational, particle number, and reflection symmetry for even-even nuclei. The restoration of rotational symmetry does not require using bases closed under rotation. Furthermore, we added the SeaLL1 functional and improved the calculation of the Coulomb potential. Finally, we refactored the code to facilitate maintenance and future developments. New version program summaryProgram title:hfbtho v4.0CPC Library link to program files:https://doi.org/10.17632/c5g2f92by3.2Code Ocean capsule:https://codeocean.com/capsule/5389629Licensing provisions: GPLv3Programming language: Fortran 2003Journal reference of previous version: R.N. Pérez, N. Schunck, R.-D. Lasseri, C. Zhang and J. Sarich, Comput. Phys. Commun. 220 (2017) 363Does the new version supersede the previous version: YesReasons for the new version: This version adds new capabilities to restore broken symmetries and determine corresponding quantum numbers of even-even nucleiSummary of revisions:1.Angular momentum projection for even-even nuclei in a deformed basis;2.Particle number projection for even-even nuclei in the quasiparticle basis;3.Implementation of the SeaLL1 functional;4.Expansion of the Coulomb potential onto Gaussians;5.MPI-parallelization of a single hfbtho execution;6.Code refactoring.Nature of problem:hfbtho is a physics computer code that is used to model the structure of the nucleus. It is an implementation of the energy density functional (EDF) approach to atomic nuclei, where the energy of the nucleus is obtained by integration over space of some phenomenological energy density, which is itself a functional of the neutron and proton intrinsic densities. In the present version of hfbtho, the energy density is derived either from the zero-range Skyrme or the finite-range Gogny effective two-body interaction between nucleons. Nuclear superfluidity is treated at the Hartree-Fock-Bogoliubov (HFB) approximation. Constraints on the nuclear shape allow probing the potential energy surface of the nucleus as needed, e.g., for the description of shape isomers or fission. A local scale transformation of the single-particle basis in which the HFB solutions are expanded provides a tool to properly compute the structure of weakly-bound nuclei. Restoration of the rotational, particle number, and reflection symmetry for even-even nuclei enables recovering the quantum numbers that are lost at the HFB approximation.Solution method: The program uses the axial harmonic oscillator (HO) or the transformed harmonic oscillator (THO) single-particle basis to expand quasiparticle wave functions. It iteratively diagonalizes the HFB Hamiltonian based on generalized Skyrme-like energy densities and zero-range pairing interactions or the finite-range Gogny force until a self-consistent solution is found. Lagrange parameters are used to impose constraints on HFB solutions, and their value is updated at each iteration from an approximation of the quasiparticle random phase approximation (QRPA) matrix. Symmetry restoration is implemented through standard projection techniques. Previous versions of the program were presented in [1-3].Additional comments including restrictions and unusual features: Axial and time-reversal symmetries are assumed in HFB calculations; y-simplex symmetry and even particle numbers are assumed in angular momentum projection.

Analysis of drag reduction effects in turbulent Taylor–Couette flow controlled via axial oscillation of inner cylinder

Analysis of drag reduction effects due to axial oscillation of an inner cylinder in a turbulent Taylor–Couette (TC) flow is performed in the present study. The frictional Reynolds number on the inner cylinder is 218, and the non-dimensional oscillating period is varied from 8 to 32. By examining turbulence statistics, we uncover different impacts of the long- and short-period oscillations on the circumferential (θ) and radial (r) velocity fluctuations in large (uθl, url) and small (uθs, urs) scales. One of the most surprising findings is that the short-period oscillation increases the large-scale Reynolds shear stress ⟨uθlurl⟩ by the strong intensification of uθl exceeding the suppression of url. To understand the phenomena, the spectra of each term in the transport equations of the Reynolds normal stresses ⟨uθ′uθ′⟩ and ⟨ur′ur′⟩ are analyzed. First, it is shown that the short-period oscillation weakens the productions of uθs, urs, and url while it enhances that of uθl. In contrast, the long-period oscillation reduces the productions of uθl and url while it mainly intensifies that of urs. Second, the investigations of the pressure–strain terms indicate that the short-period oscillation mainly impedes the inter-component energy transfer originating from the small-scale background turbulence. However, the long-period oscillation benefits the small-scale inter-component energy communication while it hinders the large-scale one. In addition, the inverse energy transfer in the turbulent TC flow is confirmed by inspecting the inter-scale energy transfer terms. The hindrance of the inter-scale energy transfer by the inner-cylinder oscillation plays a non-negligible role in the reduction of the wall friction drag.

Temporal Instability of Liquid Jet in Swirling Gas with Axial Velocity Oscillation

In the present study, viscous potential flow theory is employed to investigate the temporal instability of a viscous liquid jet in swirling gas with axial velocity oscillation. The dispersion-relation equation is obtained between dimensionless growth rate and wave number. Because of the axial velocity oscillation of gas, there are more than one instability regions, including Kelvin–Helmholtz (K–H) instability region and parametric instability region. The results suggest that increasing gas swirling strength stabilizes the liquid jet in axisymmetric mode, while it enhances the instability in nonaxisymmetric modes. And the gas viscosity has a destabilizing effect. In addition, increasing the forcing frequency enhances instability in the K–H instability region. However, it has a stabilizing effect in the parametric instability region. Furthermore, increasing oscillation amplitude enhances instability in both the K–H instability region and the parametric instability region. Because of the competition between the parametric and K–H instabilities, the location of dominant wave number is uncertain.

Improved Ultrasound Imaging Performance With Complex Cumulant Analysis.

Ultrasound localization microscopy (ULM) breaks the acoustic diffraction limit. However, the temporal resolution of ULM is relatively low because of a long data-acquisition time. Inspired by super-resolution optical fluctuation imaging (SOFI), in this paper, we propose a method for ultrasound imaging with improved imaging performance, which is achieved by using cumulant analysis. Specifically, to eliminate the axial oscillations, here, the cumulant analysis framework is extended, which is used to process the complex-valued analytic signals rather than the real-valued signals. The results from the numerical simulations and in vitro physical phantom experiments indicate that by generalizing cumulant analysis to complex-valued signals, a high imaging performance is achieved with an improvement of ∼35%-42% (lateral direction) and ∼41%-42% (axial direction) in the resolution compared with the temporal mean envelope image, in terms of full-width-half-max (FWHM). In particularly, the axial oscillations appearing in the real cumulant images are effectively eliminated by the complex cumulant analysis. Moreover, the proposed method can easily take advantage of SOFI. In the phantom experiment, a short data-acquisition time (∼2 sec) is enough to obtain the improved spatial resolution. The proposed method offers an implementation of US with high spatial resolution, fast data-acquisition speed, and axial oscillations removal characteristics. The method provides the potential in US imaging fast biological processes in vivo.

Higher order Motional Resonances Spectra of electrons with Non-linear Axial Oscillations in Quadrupole Penning trap

Experimentally the motional resonances spectra of an electron cloud confined in a quadrupole Penning trap, weakly excited with RF field, which is swept from 1 MHz to 1 GHz was examined. The higher order motional resonances of the electron cloud with non-linear axial oscillations show that they result from couplings in the motional degrees of freedom, which characterize the perturbations in the trap, and cause loss of electrons from the trap. The likely presence of perturbative terms of quadrupole potential is identified by studying the motional resonances spectra and hence the multipole terms are calculated. A periodically forced barrel spring with a steel ball at the free end can be regarded as a model of non-linear oscillator and a periodically forced steel ball hung through a flexible rod, which is deflected alternately toward the two magnets can be regarded as a mechanical model of Duffing oscillator are developed, in analogy to observed higher order motional resonances of the electron cloud excited with different RF field powers.