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This paper examines the oscillations of a spherical pendulum with horizontal Lissajous excitation. The pendulum has two degrees of freedom: a rotational angle defined in the horizontal plane and an inclination angle defined by the pendulum with respect to the vertical z axis. The results of numerical simulations are illustrated with the mathematical model in the form of multi-colored maps of the largest Lyapunov exponent. The graphical images of geometrical structures of the attractors placed on Poincaré cross sections are shown against the maps of the resolution density of the trajectory points passing through a control plane. Drawn for a steady-state, the graphical images of the trajectory of a tip mass are shown in a three-dimensional space. The obtained trajectories of the moving tip mass are referred to a constructed bifurcation diagram.

We perform an analytical study of the Lissajous and halo orbits around collinear points L1 and L2 in a spatial circular restricted three-body problem of an arbitrary value of the mass ratio. Using a canonical transformation procedure, we generate complete and resonant normal forms through reduction to center manifolds. The coefficients in the normal forms are explicitly expressed as functions of mass ratio for the first time so that one can evaluate the energy level at which bifurcation of halo orbit takes place. Another contribution of this paper is giving the analytical solutions of Lissajous and halo orbits in the initial synodic reference system through the inverse transformation of normalization. The analytical results are the series form of normalized action-angle variables, and their coefficients are also explicitly expressed as functions of mass ratio. Finally, comparison results demonstrate that the solutions for a Lissajous orbit derived through normalization method and Lindstedt–Poincare method are completely the same, while the solutions for a halo orbit derived through these two methods are different but have the roughly equal accuracy.

Research has supported two distinct forms of motor skill consolidation that can occur between practice sessions: (1) off-line learning, and (2) memory stabilization. Off-line learning describes performance improvement between practice sessions that is above the gain observed at the end of practice, while memory stabilization describes a gain in performance that is maintained between practice sessions. This study used a Lissajous plot to provide concurrent feedback to train participants to produce a 90° relative phase between the index fingers (flexion/extension motion). Significant improvements in performance emerged after ten trials (5 min) of practice. At the end of training, participants were divided into two delay interval groups before retesting, 2-h and 6-h. The retesting session started with participants performing an interference task (10 trials, 5 min) that required training on a 45° relative phase between the fingers with concurrent feedback from the Lissajous plot. When training with the interference task was completed participants were retested with the 90° relative phase without the Lissajous plot feedback. In the retest of the 90° pattern, a performance loss was found in the 2-h delay group, whereas the 6-h delay group maintained the end of practice performance level. Maintenance of the same level of performance without the Lissajous plot represents memory stabilization of the initially trained 90° pattern. The findings are discussed within the context of current positions regarding procedural consolidation and the coordination dynamics framework wherein action and perception are linked through the informational nature of relative phase.

Study of the behavior of polymer solution flows in the region of nonlinear viscoelasticity allows one to more accurately evaluate the adequacy of rheological models and to describe the rheological properties of a material in more detail. The nonlinear viscoelastic properties manifesting themselves in the process of studying the behavior of a polymer material on significant deformations were investigated with the aid of time dependences of shear stresses calculated at different amplitudes. The present work considers the applicability of the modified Vinogradov–Pokrovskii rheological model to describing the oscillating shearing of polymer fluids with a large amplitude. It has been established that on increase of the deformation amplitude, the shear stresses cease to be a true harmonic, and one observes the appearance of a "step" on their left front, which speaks of the substantial nonlinearity in the behavior of the sample. The obtained theoretical dependences are compared with experimental data for a 5% solution of polyethylene oxide in dimethyl sulfoxide. The comparison was made as by plotting the time dependences of normalized stresses, so by analyzing Lissajous figures. Despite the simplicity, the modifi ed Vinogradov–Pokrovskii rheological model adequately describes the behavior of polymer materials on significant periodic deformations.

This paper proposes a new signal transformed internal model control (STIMC) scheme for non-raster scanning patterns of piezo-actuated nanopositioning stages. To smooth the scanning signals superimposed with a ramp or time-varying amplitudes, a signal transformation operator is calculated to convert the references into standard harmonic waveforms. An inverse transformation operator is then added in the control loop to generate the driving signals. For the contained internal model control (IMC) design, an H∞ mixed-sensitivity method is utilized for the first order internal mode synthesis. A second and a third internal modes are included in the IMC for alleviating residual high-frequency errors resulted from the nonlinearity of hysteresis. To verify the proposed STIMC scheme, comparative experiments with conventional IMC are conducted based on a nanopositioning platform. Results prove that the STIMC is effective on non-raster signals’ tracking. A same tracking precision for Lissajous scanning can be obtained by STIMC compared with IMC. An improvement of larger than 50% and 80% of root-mean-square errors can be obtained for cycloid and spiral scanning patterns, respectively.

Both intrapersonal and interpersonal coordination dynamics have traditionally been investigated using relative phase patterns of in-phase (ϕ = 0°) and/or anti-phase (ϕ = 180°). Numerous investigations have demonstrated that coordination tasks that require other relative phase patterns (e.g., 90°) are difficult or near impossible to perform without extended practice. Recent findings, however, have demonstrated that an individual can produce a wide range of intrapersonal bimanual patterns within a few minutes of practice when provided integrated feedback. The present experiment was designed to directly compare intra- and interpersonal coordination performance and variability when provided Lissajous feedback or pacing metronome. Single participants (N = 12) and pairs of participants (N = 24, 12 pairs) were required to produce relative phase patterns between 0° and 180° in 30° increments using either pacing metronomes or Lissajous displays. The Lissajous displays involved a goal template and a cursor providing integrated feedback regarding the position of the two effectors. The results indicated both single and pairs of participants could effectively produce a large range of coordination patterns that typically act as repellers after only 6 min of practice when provided integrated feedback. However, single participants performed the in-phase coordination pattern more accurately and with less variability than paired participants, regardless of the feedback condition. These results suggest an advantage for intrapersonal coordination when performing in-phase coordination, possibly due to the stabilizing effect occurring via the neuro-muscular linkage between effectors.

Lissajous curves have been used in various engineering applications such as optics, imaging, antenna scan, machining, as well as mobile robotics. In this article, we propose and analytically justify a Lissajous curve based trajectory planning strategy for aerial multi-agent systems to achieve the following objectives simultaneously: (i) Collision free paths for repeated coverage of a region while maintaining a closed sensor ring around a specified center for all time. (ii) Guaranteed detection of any stationary or moving object enclosed within the ring in finite time without the possibility of undetected escape. This leverages known and some novel properties of Lissajous curves that we establish as a part of this work. This has several potential applications in civil and military missions such as search and surveillance, repeated patrolling, target detection and capture, and the proposed strategy meets all these objectives simultaneously. We validate the proposed strategy through simulations and experiments using differential drive ground robots. We also demonstrate the applicability of this strategy for aerial surveillance through a Software-In-Loop-Simulation for quadrotors.

Micro series resistance spot welding (MSRSW) is a common practice to form two neighboring nuggets at the same time in industrial fabrication of precision parts, especially in EV battery pack manufacturing. However, because of the Peltier effect, unbalanced Joule heat generation results in different nugget sizes. Therefore, weld quality monitoring systems are necessary to improve weld quality. Unfortunately, no work has been reported. It is still a challenge to develop a monitoring system. A series of monitoring systems have been developed for conventional resistance spot welding (RSW). Among them, electrode movement caused by heat expansion and contraction during the welding process is an important information. GAP sensor and LVDT are major sensors used in the measurement of electrode movement. However, the physical dimension of these sensors is too huge to be applied in MSRSW. This study proposes an in-process electrode movement monitoring system for MSRSW using dual accelerometers. Two-minute accelerometers were mounted on top of each electrode holders. Thanks to its good sensitivity and frequency response characteristics, an accelerometer signal was successfully picked up and integrated twice to reflect an electrode expansion/contraction movement. The correlation between two signals was analyzed by applying the Lissajous figure technique. Experimental verification of this monitoring system has been carried out. A major finding is that dual accelerometer signal and its double-integrated signal, electrode movement, reflect thermo-physical phenomena during MSRSW process.

In this work, the FENE dumbbell model under small- and large-amplitude oscillatory shear flows using a micro-macro approach is presented. This approach involves the evolution of an ensemble of Brownian Configuration Fields which describes the polymer dynamics of the microscopic scale and the momentum equation describes the macroscopic scale. The Lissajous curves for the shear stress and the first normal stress difference versus the instantaneous strain or strain rate for the elastic or viscous projection are shown. The influences of the solvent/polymer viscosity ratio, the maximum extension length, and the relation between strain rate and frequency are analyzed. An important finding is the self-intersection of the Lissajous curves, which forms secondary loops for short extension lengths and high Weissenberg/Deborah dimensionless numbers ratio.

In this work, we propose a number of systematic experiments performed on series LCR-type linear resonant circuits driven by a square wave input voltage source to generate beautiful images of periodic attractors on the screen of an oscilloscope by altering circuit’s elemental order. Besides, we also indicate how such simple linear circuits can be extended to perform numerical simulations by incorporating the mathematical of logarithmic spiral. The numerical simulation outcomes are in agreement with the laboratory results. This work could be useful in demonstrations of phase flow in a linear dynamics/nonlinear dynamics course and will pave the way to easier understand papers in nonlinear dynamics of nonlinear electronic circuits with both experimental and numerical results like Grosu et al. (Phys Rev Lett 100(23):234102, 2008). To the best of our knowledge, this work is innovative of its kind giving a prescription to present an exhibition of a rich variety of periodic attractors. We also propose that the present work has the potential to be used as classroom demonstration in topics like resonant circuits, damped harmonic oscillators, oscilloscopes, Lissajous figures, phase portraits, periodic attractors or an exciting experiment in the advance level physics and engineering laboratories utilizing such simple circuitry.

The bicircular model is a periodic time-dependent perturbation of the Earth–Moon restricted three-body problem that includes the direct gravitational effect of the Sun on the infinitesimal particle. In this paper, we focus on the dynamics in the neighbourhood of the $$L_1$$ point of the Earth–Moon system. By means of a periodic time-dependent reduction to the centre manifold, we show the existence of two families of quasi-periodic Lyapunov orbits, one planar and one vertical. The planar Lyapunov family undergoes a (quasi-periodic) pitchfork bifurcation giving rise to two families of quasi-periodic halo orbits. Between them, there is a family of Lissajous quasi-periodic orbits, with three basic frequencies.

This work focuses on the dynamic rheological behavior of low water-to-binder ratio cement mortars blended with fly ash microspheres (FAM) or silica fume (SF). The initial slump flow of each group has been controlled at similar values by adjusting the superplasticizer dosages. With the help of a coaxial cylinder rheometer, the dynamic rheological behaviors of these mortars are investigated by frequency sweeping in the range of 0–2 Hz under large amplitude oscillatory shear (LAOS). Based on the systematical elaboration of dynamic rheological testing theory, the experimental data are processed according to Lissajous plot fitting to reveal the viscoelastic characteristics. The nonlinearity of response signals is further assessed with Fourier transform (FT) analysis. The parameters, storage modulus G', loss modulus G" and relative amplitude I3/I1 are proposed to clarify the influences of FAM and SF on the stability and energy consumption of local structures and nonlinearity of response torques. The hydration characteristics of various groups well confirmed the rheological phenomenon. This study is beneficial for the preparation and optimization of flow state concrete such as pumping concrete and self-compacting concrete.

International audience; The use of granular material is particularly attractive to damp vibrations in hollow structures, due to grains relative motion that induce shocks and friction, interaction grains/structure and possible deformation of the grains. Therefore, granular dampers are of great interest for many application types.However, predictive modelling of the damping performance of a granular material remains particularly difficult. The objective of this experimental work is to evaluate the vibration damping performance of various granular material samples, when inserted in a vertically shaken rigid cavity. This is done by Lissajous' representation of displacement and force time signals, which, applied to this type of system, is a methodological originality allowing a detailed analysis of the overall motion of the grain cluster. The main insights are the identification of links between certain control parameters of the granular material and the vibratory energy dissipated. From Lissajous' representation post-processing, it is shown that dissipation increases linearly with grain mass; at given mass the dissipation increases with grain size; the use of a viscoelastic material notably increases the performance, particularly at low acceleration levels; that the roughness of surfaces in contact with the grains plays a secondary role. Finally, the set of parametric variations give a better physical understanding of the phenomena involved, which gives orientations for future modelling works.

We propose a new concept of single-shot deflectometry for real-time measurement of three-dimensional surface profile using a single composite pattern. To retrieve an accurate phase from one-frame composite pattern, we adapt the Fourier Transform (FT) method and the spatial carrier-frequency phase-shifting (SCPS) technique to our proposed deflectometry. Based on Lissajous figure and ellipse fitting method, we also correct the phase extraction error in SCPS technique by reducing the effect of background and modulation variations. The proposed technique is verified by comparing our measurement results with phase-shifting deflectometry, and the maximum difference between two measurement results is less than 30 nm rms. We also test the robustness to vibration and the measurement capability for dynamic object.

An attempt was made to study the dynamics close to the collinear libration point $$ L_{2} $$ of the radial solar sail circular-restricted three-body problem (RSCRTBP) in the Sun–Jupiter System, where the third massless body is a solar sail. We analyse the qausi-periodic (Lissajous solutions) orbits about the libration point $$ L_{2} $$ . The Lindstedt–Poincare approximation for the qausi-periodic orbits was used for numerical simulations. We utilized linear quadratic regulator (LQR) to stabilize the full nonlinear model, and linear state-feedback controller was designed to stabilize the trajectory.

The ability to coordinate the simultaneous movements of our arms is limited by a coalition of constraints. Some of these constraints can be overcome when the task conceptualisation is improved. The present study investigated how the movement preparation of bimanual reaching movements was affected by integrated visual feedback of the responses. Previous research has shown that the preparation of bimanual asymmetric movements takes longer than bimanual symmetric movements. The goal of the present study was to determine whether integrated, Lissajous feedback could eliminate this bimanual asymmetric cost. Fifteen participants made unimanual and bimanual symmetric and asymmetric reaches with separate feedback, where there was a cursor and a target for each hand. Participants also made bimanual symmetric and asymmetric movements with integrated feedback; a single cursor and a single target represented the locations and goals of both arms in this condition. The results showed a bimanual asymmetric cost with separate feedback, and that this cost persisted with integrated feedback. We suggest that integrated feedback improved continuous and discrete bimanual movements in other experiments by facilitating error detection and correction processes. We hypothesise that the bimanual asymmetric cost persisted in the present experiment because the uncertainty associated with choice reaction time prevented the facilitated error processing from improving the preparation of the next trial.

Dough blended with rocket leaves powder was subjected to small and large amplitude oscillatory shears. Small amplitude oscillatory shear data were fitted to a discrete relaxation model of elastic solids and to a critical gel model. The small amplitude relaxation spectrum was thereafter used to calculate the LAOS predictions of various large deformation models. The LAOS theoretical calculations using the Phan-Thien model showed good agreement with the first harmonic stress data, and only qualitative agreement with the third and the fifth harmonic stress values. Lissajous curves showed dissimilarity in shape between the experimental data and Phan-Thien model. The network model of Sim et al. (2003). Did not have the butterfly shape displayed in the Phan-Thien model, but it provided a worse fit to stress harmonics than the Phan-Thien model. An improved damage function was proposed, where time effect on network damage was taken into consideration, and fits to stress harmonics and to Lissajous stress-strain curves were significantly improved.

Interfacial rheological measurements often show in their nonlinear Lissajous-plots rhombus or saddle-like shapes indicating complex local deformation behaviour. A strong interacting silica particle and an almost not interacting clay particle were studied in respect to their interfacial rheological properties. Large amplitude oscillation shear measurements were performed with a bicone geometry and combined with optical measurements, from which particle tracks were calculated. A correlation was found between the appearance of shear localisation and Lissajous-plot shapes. Silica particles showed shear localisation at the bicone edge and rhombic plateaus in the Lissajous-plot, while the shear localisation for the clay particles was observed at the cup's wall as saddle-like shaped Lissajous-plots.

Action-angle variables for the Levi-Civita regularized planar Kepler problem were introduced independently first by Chenciner and then by Deprit and Williams. The latter used explicitly the so-called Lissajous variables. When applied to the transformed Keplerian Hamiltonian, the Lissajous transformation encounters the difficulty of being defined in terms of the constant frequency parameter, whereas the Kepler problem transformed into a harmonic oscillator involves the frequency as a function of an energy-related canonical variable. A simple canonical transformation is proposed as a remedy for this inconvenience. The problem is circumvented by adding to the physical time a correcting term, which occurs to be a generalized Kepler’s equation. Unlike previous versions, the transformation is symplectic in the extended phase space and allows the treatment of time-dependent perturbations. The relation of the extended Lissajous–Levi-Civita variables to the classical Delaunay angles and actions is given, and it turns out to be a straightforward generalization of the results published by Deprit and Williams.

Dynamic oscillatory measurement, i.e., small amplitude oscillatory shear (SAOS) and large amplitude oscillatory shear (LAOS) test was used to investigate linear and non-linear viscoelastic properties of Polypropylene (PP)/Polystyrene (PS) blends with and without 5 wt.% clay (C20A). Fourier transform (FT-Rheology), Lissajous curves and stress decomposition methods were used to analyze non-linear responses under LAOS flow. Composition effects of blends were investigated prior to compatibilization effects. Elevated concentrations of dispersed phase (PS) increased the moduli G'(ω) from SAOS test and G*(γ 0) from LAOS test of the blends as well as strain thinning behavior. Interestingly, addition of 5 wt.% clay (C20A) boosted moduli of the blends as well as led to similar strain thinning behaviors among the PP/PS/C20A blends, except for the (90/10) PP/PS blend. The latter did not show improved rheological properties despite morphological improvements, as shown by SEM. Results from SEM and improved rheological properties of PP/PS/C20A blends revealed the compatibilization effects of clay (C20A) particles regardless of size reduction mechanisms. Third relative intensities (I 3/1) from FT-rheology were found to be sensitive to clay (C20A) additions for the (70/30) and (30/70) PP/PS blends. Similarly, Lissajous curves could detect changes upon clay (C20A) addition, specifically at lower strain amplitudes whereupon addition of 5 wt.% clay resulted in the closed loops of Lissajous curves showing a more ellipsoidal shape due to increased elasticity in the blends. However, detection of these changes at larger strain amplitudes was more challenging. Therefore, stress decomposition (SD) method was applied for more precise characterization as it decomposes the total stress (σ) into elastic stress (σ′) and viscous stress (σ″). Using SD method, elastic stress was more distorted, especially, strain hardening, while the total stress response remained almost unchanged at larger strain amplitudes.

We study the dynamics near the collinear Lagrangian points of the spatial, circular, restricted three-body problem. Following a standard procedure, we reduce the system to the center manifold and we analyze the Lissajous orbits as well as the halo orbits, the latter ones arising from bifurcations of the planar Lyapunov family of periodic orbits. To obtain the Lissajous orbits, we perform a classical perturbation theory and we provide a formal approximate solution under suitable non-degeneracy and non-resonance conditions. As for the halo orbits, we construct a normal form adapted to the synchronous resonance: introducing a detuning, measuring the displacement from the resonance, and expanding the energy in series of the detuning, we are able to evaluate the energy level at which the bifurcation takes place. Except for a particular case, the analytical values obtained after a second order resonant perturbation theory are in very good agreement (in some cases up to the fourth decimal digit) with the numerical values found in the literature.

A class of random stationary, scalar sources producing far fields with Lissajous-like intensity profiles is introduced by modeling the source spectral degree of coherence (SDOC) as a multi-cosine Gaussian-correlated Schell-model (McGCSM) function. Analytical expressions for the cross-spectral density generated by such sources on propagation through a stigmatic ABCD optical system are derived. The far-field spectral density distributions in free space are illustrated numerically. It is shown that the McGCSM beam can exhibit Lissajous-like patterns during propagation when the beam parameter M is sufficiently large. The patterns can be controlled to demonstrate various geometries by directly adjusting the properties of its SDOC in the source plane. Moreover, we find that the far-field spectral density distribution can also display an optical lattice structure when the beam parameter M is relatively small.

A new method of obtaining bounded trajectories near collinear-like equilibrium points of elongated asteroids using a simple linear control is proposed. The characteristic equation of the linearized equation of motion near the collinear-like equilibrium points is reduced based on the property of the second derivatives of the effective potential. According to the reduced characteristic equation, the motion is decoupled into an in-plane motion and a out-of-plane motion. The solutions to the decoupled equation of motion are derived and the corresponding Lissajous trajectories are chosen as desired trajectories. A simple linear control, which tracks the Lissajous trajectories, is formulated with the stability conditions of gains. The effectiveness of the proposed method is validated by simulations using high fidelity models of 216 Kleopatra and 951 Gaspra.

This paper presents the design and construction of non-thermal plasma jet device which was built in plasma phys. Dept., NRC, AEA, Egypt with a plasma application group. This design will be useful to initiate research in different fields such as low temperature plasma, polymer and biomedical applications. The experimental operation of this device is conducted with power supply of (10 kV, 30 mA, and 20 kHz). The discharge process takes place by using Air as input gas with different flow rates. The experimental results showed that the maximum plasma jet length of 7 mm is detected at air flow rate of 12 L/min. The electrical characteristics of discharge at different flow rates of Air such as discharge voltage, current, mean power, power efficiency, and energy have been studied by using potential dividers and Lissajous figure techniques. The results of plasma jet temperature along the jet length showed that the jet plasma has approximately a room temperature at the end of jet column.

Microelectromechanical systems fabrication and packaging may induce parametric defects which have to be efficiently diagnosed to improve fabrication yield and device reliability. In this work, the possibilities of diagnosing single parametric defects in dual axis IC accelerometers are explored. The proposed method is a reuse of a mixed-signal diagnosis strategy based on Lissajous compositions. It consists of a dynamic correlation of the two orthogonal output signals of the device. This leads to a Lissajous trace which is able to manifest parametric deviations. Several device parameters have been studied and characterized under different deviation levels. The definition of a metric and the evaluation of its diagnosis efficiency is used to optimize the diagnosis procedure against noisy measurements. The method has been experimentally applied to diagnose the misalignment degree in a commercial dual axis accelerometer. The Lissajous characterization of the defect is illustrated and experimental results are reported. Misalignment diagnosis results show the viability of the proposal.

This investigation explores regions near libration points that might prove suitable for space observations with large formations. Recent analyses have considered occulters located at relatively large distances from the telescope near the L 2 Sun-Earth libration point for detection of exoplanets. During the science mode, the telescope-occulter distance, as well as the pointing direction toward the star, are typically fixed. Quasi-periodic Lissajous trajectories are employed as a tool to determine regions near the telescope orbit where the large formation can be maintained. By placing the occulter in these locations, the control required to maintain the line-of-sight is reduced.

Background Thoracoabdominal asynchrony is often adopted to discriminate respiratory diseases in clinics. Conventionally, Lissajous figure analysis is the most frequently used estimation of the phase difference in thoracoabdominal asynchrony. However, the temporal resolution of the produced results is low and the estimation error increases when the signals are not sinusoidal. Other previous studies have reported time-domain procedures with the use of band-pass filters for phase-angle estimation. Nevertheless, the band-pass filters need calibration for phase delay elimination. Methods To improve the estimation, we propose a novel method (named as instantaneous phase difference) that is based on complementary ensemble empirical mode decomposition for estimating the instantaneous phase relation between measured thoracic wall movement and abdominal wall movement. To validate the proposed method, experiments on simulated time series and human-subject respiratory data with two breathing types (i.e., thoracic breathing and abdominal breathing) were conducted. Latest version of Lissajous figure analysis and automatic phase estimation procedure were compared. Results The simulation results show that the standard deviations of the proposed method were lower than those of two other conventional methods. The proposed method performed more accurately than the two conventional methods. For the human-subject respiratory data, the results of the proposed method are in line with those in the literature, and the correlation analysis result reveals that they were positively correlated with the results generated by the two conventional methods. Furthermore, the standard deviation of the proposed method was also the smallest. Conclusions To summarize, this study proposes a novel method for estimating instantaneous phase differences. According to the findings from both the simulation and human-subject data, our approach was demonstrated to be effective. The method offers the following advantages: (1) improves the temporal resolution, (2) does not introduce a phase delay, (3) works with non-sinusoidal signals, (4) provides quantitative phase estimation without estimating the embedded frequency of breathing signals, and (5) works without calibrated measurements. The results demonstrate a higher temporal resolution of the phase difference estimation for the evaluation of thoracoabdominal asynchrony.

We present a mathematical model for animating trees realistically by taking into account the influence of natural frequencies and damping ratios. To create realistic motion of branches, we choose three basic mode shapes from the modal analysis of a curved beam, and combine them with a driven harmonic oscillator to approximate Lissajous curve which is observed in pull-and-release test of real trees. The forced vibration of trees is animated by utilizing local coordinate transformation before applying the forced vibration model of curved beams. In addition, we assume petioles are flexible to create natural motion of leaves. A wind field is generated by three-dimensional 1/f β noises to interact with the trees. Besides, our animation model allows users to interactively manipulate trees. We demonstrate several examples to show the realistic motion of interactive trees without using pre-computation or GPU acceleration. Various motions of trees can be achieved by choosing different combinations of natural frequencies and damping ratios according to tree species and seasons.

An adaptive digital demodulation method for a sinusoidal encoder signal is proposed for fast and accurate analysis of spiral servo track writing (SSTW). The proposed methods combine adaptive signal correction and digital interpolation. In addition, digital phase-shifted sinusoid (PSS) interpolation is proposed in order to acquire precise position information. The adaptive signal correction method compensates for a fast-motion non-ideal encoder signal. The digital PSS can be simply implemented and can resolve the switching noise of conventional analog methods. The proposed adaptive digital demodulation scheme can be simply digitalized and can improve the position control in SSTW. From the simulation and experimental results, we have validated the proposed methods, which show noticeable improvement in terms of the Lissajous error and the position error signal.

The non-symmetric responses of normal stresses in oscillatory squeeze flow have been investigated with model calculations. The simplest and most widely used constitutive equations were employed to predict the non-symmetric normal stresses, which is a distinctive feature of oscillatory squeeze flow. The model prediction was compared with experimental data of polymer solution in terms of stress shape, Lissajous plot, stress decomposition, and Fourier transformation. The upper-convected Maxwell, Giesekus, and exponential Phan-Thien Tanner models predicted the nonsymmetric characteristics of normal stresses under oscillatory squeeze flow. The predictions showed fairly good agreement with experimental data. However, the upper-convected Maxwell model showed unrealistic result in the Lissajous plot of [stress vs. strain] and [stress vs. strain rate]. From stress decomposition, it could be confirmed that the non-symmetric nature arises from the elastic contribution of the normal stress, which was verified in both experiment and model calculation. This study is expected to provide useful insights for further understanding of the nonlinear and non-symmetric characteristics of oscillatory squeeze flow.

Large Amplitude Oscillatory Shear (LAOS) is a test method for the characterization of complex fluids. Varying independently both strain amplitude (γ0) and frequency (ω) allows covering a broad spectrum of rheological responses with respect to time scales and involved non-linearity. Moreover, it is experimentally relatively simple to generate LAOS flow, because dynamic oscillatory shear does not involve any sudden jump in either strain or strain rate. There are several methods to analyze the resulting torque data received from the LAOS test protocol: (1) the G′ and G″ as a function of strain amplitude (2) Stress shape (stress vs. time) or Lissajous pattern (stress vs. strain or stress vs. strain rate) (3) Fourier transform (4) generalized “storage” and “loss” modulus when decomposing the nonlinear stress data (5) Chebyschev polynomials using decomposing stress data and further development of Chebyschev polynomials. The Fourier Transform (FT)-Rheology is perhaps the most sensitive method of those discussed above. A new nonlinear parameter Q established from FT-Rheolgy under LAOS flow, i.e.Q(ω,γ0) ≡ I3/1/γ02, as well as the zero-strain nonlinearity or intrinsic nonlinearity \(Q_0 \left( \omega \right) \equiv \lim _{\gamma _0 \to 0} Q\left( {\omega ,\gamma _0 } \right)\) by Hyun and Wilhelm (2009). In this study, therefore recent experiment and simulation results of nonlinear parameter Q from FT-Rheology for polymer melt and polymer composite systemsare reviewed.

An experiment was conducted to determine whether multi-frequency continuous bimanual circling movements of varying difficulty (1:2, 2:3, 3:4, and 4:5) could be effectively performed following relatively little practice when on-line continuous relative velocity feedback is provided. The between-subjects results indicate extremely effective bimanual multi-frequency performance for all coordination patterns with relatively stable and continuous movements of both limbs. The findings suggest that the previous performance effects using Lissajous feedback with reciprocal movement can be extended to circling movements using on-line relative velocity feedback. Contrary to the long-held position that these coordination patterns result in increasing difficulty, we failed to find systematic relative velocity error, variability, or bias differences between the participants performing the various multi-frequency coordination patterns. Indeed, coordination error, variability, and biases were remarkably low for each of the tasks. The results clearly indicate the ease with which participants are able to produce bimanual coordination patterns typically considered difficult if not impossible when salient visual information is provided that allows the participants to detect and correct their coordination errors.

Except for the Lissajous singularities, the singular Lissajous line is shown to be in existence, which is emergent from the incoherent addition of two uncorrelated vector electric fields at arbitrarily different frequencies. The free-space propagation of the emergent Lissajous singularities and singular Lissajous line and their interaction are studied theoretically and illustrated numerically.

We present a fast, displacement-measuring, single-pass, two-detector homodyne quadrature laser interferometer and compare its performance with an arm-compensated, proportional, integral-derivative-controlled Michelson interferometer. Special attention is given to the extension of the dynamic range. The wide dynamic range is achieved by an accurate fringe subdivision based on an enhanced ellipse-specific fitting of the scattered Lissajous curve and by increasing the total displacement using the quadrature-detection technique. The common periodic deviations, i.e., the unequal AC amplitudes, the DC offsets, and the lack of quadrature are determined and reduced by data processing based on an ellipse-specific, least-squares fitting to obtain nanometric accuracy. The performance of the described interferometer is demonstrated through the measurement of high-amplitude and high-frequency laser-induced ultrasound.

Due to various perturbations, the collinear libration points of the real Earth–Moon system are not equilibrium points anymore. Under the assumption that the Moon’s motion is quasi-periodic, special quasi-periodic orbits called dynamical substitutes exist. These dynamical substitutes replace the geometrical collinear libration points as time-varying equilibrium points. In the paper, the dynamical substitutes of the three collinear libration points in the real Earth–Moon system are computed. For the points L1 and L2, linearized motions around the dynamical substitutes are described, and the variational equations of the dynamical substitutes are reduced to a form with a near constant coefficient matrix. Then higher order analytical formulae of the central manifolds are constructed. Using these analytical solutions as initial seeds, Lissajous orbits and halo orbits are computed with numerical algorithms.

The measurements of electro-optical discharge characteristics and concentration of produced ozone were performed to evaluate the efficiency of ozone production in an AC surface dielectric barrier discharge (SDBD) in pure oxygen at atmospheric pressure. The discharge was driven in an amplitude-modulated regime with a driving AC frequency of 1 kHz, variable discharge duty cycle of 0.01–0.8 and oxygen flow rate of 2.5–10 slm. We observed asymmetric SDBD behaviour as evidenced by the variation in the ratio of the OI/O2+ emission intensities registered during the positive/negative half-periods and complemented by the transferred charge measurements through the Lissajous figures. We also found a strong dependence of O3 concentration on the discharge duty cycle. The highest calculated ozone production yield reached 170 g/kWh with a corresponding energy cost of about 10 eV/molecule when combining the lowest inspected duty cycle with the lowest AC high voltage amplitude.

This paper presents a robust tracking control strategy using an adaptive sliding mode approach for MEMS triaxial angular sensor device that is able to detect rotation in three orthogonal axes, using a single vibrating mass. An adaptive sliding mode controller with proportional and integral sliding surface is developed and the stability of the closed-loop system can be guaranteed with the proposed adaptive sliding mode control strategy. The proposed adaptive sliding mode controller updates estimates of all stiffness errors, damping, and input rotation parameters in real time, removing the need for any offline calibration stages. To enable all unknown parameter estimates to converge to their true values, the necessary model trajectory is shown to be a three-dimensional Lissajous pattern. The numerical simulation for a MEMS triaxial angular velocity sensor is investigated to verify the effectiveness of the proposed adaptive sliding mode control scheme.

This work proposes a Lunar Global Positioning System (LGPS) and a Lunar Global Communication System (LGCS) using two constellations of satellites on Lissajous trajectories around the collinear L 1 and L 2 libration points in the Earth–Moon system. This solution is compared against a Walker constellation around the Moon similar to the one used for the Global Positioning System (GPS) on the Earth to evaluate the main differences between the two cases and the advantages of adopting the Lissajous constellations. The problem is first studied using the Circular Restricted Three Body Problem to find out its main features. The study is then repeated with higher fidelity using a four-body model and higher-order reference trajectories to simulate the Earth-Moon-spacecraft dynamics more accurately. The LGPS performance is evaluated for both on-ground and in-flight users, and a visibility study for the LGCS is used to check that communication between opposite sides of the Moon is possible. The total ΔV required for the transfer trajectories from the Earth to the constellations and the trajectory control is calculated. Finally, the estimated propellant consumption and the total number of satellites for the Walker constellation and the Lissajous constellations is used as a performance index to compare the two proposed solutions.

Various techniques have been proposed to characterize the behavior in the non-linear regime. A new theoretical framework, as proposed recently by Ewoldt et al. (J Rheol 52(6):1427–1458, 2008), provides a quantitative analysis of Lissajous figures during large-amplitude oscillatory shear (LAOS). Intra- and intercycle non-linearities, strain stiffening and softening, and shear thinning and thickening are described and can be distinguished. The new LAOS framework from Ewoldt et al. has been extended to a sinusoidal stress input. Measurements on two different samples reveal significant different results for sinusoidal strain or sinusoidal stress input. For both sinusoidal inputs, the results have been verified by cyclic stress and strain loading tests. The sinusoidal input tests are analyzed as an oscillatory test by the rheometer software and firmware, whereas the cyclic loading tests are purely rotational tests. Since both types of testing give the same results, any instrumental artifacts can be excluded. This implies that complex fluids can behave differently whether periodic stress or strain input functions outside the linear visco-elastic range are applied. All tests in controlled strain and stress in rotational and oscillatory modes have been performed with the same rheometer based on an air bearing-supported electrically commutated synchronous motor.

When the shear stress measured in large amplitude oscillatory shear (LAOS) deformation is represented as a 2-D Lissajous–Bowditch curve, the corresponding trajectory can appear to self-intersect and form secondary loops. This self-intersection is a general consequence of a strongly nonlinear material response to the imposed oscillatory forcing and can be observed for various material systems and constitutive models. We derive the mathematical criteria for the formation of secondary loops, quantify the location of the apparent intersection, and furthermore suggest a qualitative physical understanding for the associated nonlinear material behavior. We show that when secondary loops appear in the viscous projection of the stress response (the 2-D plot of stress vs. strain rate), they are best interpreted by understanding the corresponding elastic response (the 2-D projection of stress vs. strain). The analysis shows clearly that sufficiently strong elastic nonlinearity is required to observe secondary loops on the conjugate viscous projection. Such a strong elastic nonlinearity physically corresponds to a nonlinear viscoelastic shear stress overshoot in which existing stress is unloaded more quickly than new deformation is accumulated. This general understanding of secondary loops in LAOS flows can be applied to various molecular configurations and microstructures such as polymer solutions, polymer melts, soft glassy materials, and other structured fluids.

The present findings demonstrate that when participants are provided a Lissajous display with cursor indicating the position of the limbs and a template illustrating the desired movement pattern they can rapidly (10 min) and effectively (continuous relative phase errors and variability ~10 degrees ) tune in a difficult 5:3 bimanual coordination pattern and without additional practice re-tune their responding to an equally difficult 4:3 coordination pattern. The findings indicate the extreme difficulty associated with producing complex polyrhythms in previous experiments has been due to split attention when Lissajous feedback has been provided and inability of the participant to detect and correct coordination errors when only provided vision of the limbs. Effective transfer to the 4:3 polyrhythm without previous practice suggests that the perception-action system's capabilities are extensive. The present findings when viewed in the context of recent experiments using similar protocols suggest that much, but not all, of the difficulty associated with producing a variety of bimanual coordination tasks should be viewed in terms of perceptual constraints imposed by the testing environment.

We study dihedral Galois coverings arising from Lissajous’s curves by using Galois points. For any Lissajous’s curves, we can obtain dihedral Galois coverings. Indeed, if a Lissajous’s curve is closed, we find two Galois points such that its Galois group is isomorphic to the dihedral group. On the contrary, if a Lissajous’s curve is open, by taking a suitable double covering, we obtain the dihedral Galois covering. In particular, we can prove that the double covering is obtained as a space Lissajous’s curve.

The purpose of the present experiment was to observe the performance of participants attempting to produce a 1:1 bimanual coordination pattern with 90° relative phase between the arms when feedback concerning the movement of the two limbs was integrated within a Lissajous plot and when this information was withdrawn. One group was paced with an auditory metronome and the other was encouraged to increase frequency when they fell below the goal frequency. We predicted that providing a salient integrated feedback display without a metronome would allow participants to effectively tune-in the goal relative phase pattern within several minutes; instead of several days as typically found in the literature when the metronome was used. The data indicated remarkably effective performances after 5 min of practice when the metronome was not used, with motion of both limbs harmonic in nature, and continuous relative phase errors (~10°) and standard deviation of continuous relative phase (~10°) relatively small. This seems remarkable given that this coordination pattern has proven relatively difficult to perform under normal and Lissajous feedback conditions even after several days of practice. As predicted relative phase errors and variability increased substantially when the metronome was used. When the extrinsic feedback was withdrawn all participants tended to drift from the required 90° relative phase, but the cycle duration variability in the two limbs remained stable and limb motion remained harmonic in nature. The current findings suggest that some of the difficulty typically associated with producing various relative phase patterns is due to the less than optimal perceptual information available in the various testing situations and the use of pacing metronomes.

Unstable orbits about the collinear libration points require stationkeeping maneuvers to maintain the nominal path. A new method for stationkeeping such unstable orbits is proposed here using continuous thrust. The stationkeeping challenge is formulated as a nonlinear optimal control problem in the framework of the circular restricted three-body problem with the Sun and Earth/Moon center of mass as the two primaries. A recently developed control technique known as the “θ-D controller” is employed to provide a closed-form suboptimal feedback solution to this nonlinear control problem. In this approach an approximate solution to the Hamiltonian-Jacobi-Bellman (HJB) equation is found by adding perturbation terms to the cost function. The controller is designed such that the actual (numerically integrated) trajectory tracks a predetermined Lissajous reference orbit with good accuracy. Numerical results employing this method demonstrate the potential of this approach with stationkeeping costs varying between 0.52 ∼ 0.68 m/s per year (the range depending on the particular simulation parameters used), which is of the same order of magnitude as other methods using discrete maneuvers with halo orbits. The costs are modest and the method provides flexibility in selecting the “tightness” of the control versus fuel consumption. The algorithm is well-suited for integration with onboard flight software, as the nonlinear optimal control law is solved in closed-form and the Riccati equation must only be solved once, resulting in a computationally efficient controller.

Recent investigations of the effects of environmental conditions on the vibration characteristics of automotive composite materials have led to the extensive modification of an existing forced-vibration technique. Resonant response of the specimen is now measured by using noncontacting eddy-current probes instead of strain gages, and the method of clamping the specimen has been improved. Excitation-response data is obtained in digital form rather than by photographing Lissajous patterns. The errors introduced by new and old techniques and representative data for E-glasslpolyester automotive composite materials are discussed.

We report a simple technique for the dynamic optical trapping of colloidal particles. The main feature of this technique is an addition of a scanning mirror on a vibrating membrane in the optical train of a typical optical tweezers setup. When the membrane is vibrated using sound input from a normal speaker, the laser beam that is reflected off the mirror traces out a wide variety of laser patterns of Lissajous figures depending on the input sound frequency and amplitude. The resultant pattern is subsequently focused down by a laser tweezers setup onto an aqueous suspension of monodispersed polystyrene microspheres. As a result, a wide variety of planar assemblies of the colloidal microspheres corresponding to the laser patterns are organized. In addition to the optical trapping force, the scanning laser also exerts a driving force on the microspheres. Optically assisted assemblies of microspheres can hence be manipulated to translate along the laser scanning pattern. In an elliptically patterned optical trap, repetitive scanning of the focused laser beam over an assembly of optically trapped microspheres drives the microspheres into rotation.

Testing mixed-signal circuits is a difficult task due to defect modeling challenges, observability and controllability restrictions and ATE bandwidth limitations. In this paper, the X-Y Zoning test of a Biquad filter is addressed to select the optimal excitation frequency and the best partition of the X-Y plane. Thus we obtain the best sensitivity of the BIST scheme to parametric shifts of the parameters defining the filter. The study has been particularized to shifts in the natural frequency f0 of the Biquad filter. Analytical results on the best input as well as the best partition of the observed X-Y Lissajous plots are obtained. Extensive MATLAB simulations validate the proposal, which has also been validated experimentally. For these experiments, multiple implementations of the Biquad with nominal and shifted parameters have been performed using a commercial Field Programmable Analog Array (FPAA). The experimental measures show good correlation with the analytical expressions and the simulations performed, and validate the proposed testing methodology.

This paper describes a new digital signal processor (DSP) imbalance measurement system dedicated to real-time vibration analysis on rotating machine. To accomplish real-time analysis, the vibration signals are on-line acquired and processed to analyze the mass imbalance and phase position. This is achieved through the use of FFT and Lissajous diagram. The method followed to analyze the mass imbalance with the chosen hardware and software solutions are described in detail in this paper. Several experimental tests demonstrate the efficiency and accuracy in imbalance analysis performance of the DSP system.

A complete methodology for the transfer between two Lissajous orbits around the same collinear equilibrium point is developed making use of the geometry of the phase space around these points. The result of the study gives both strategies to change the amplitudes of the Lissajous orbit and simple procedures to avoid the exclusion zone in an optimal way. The presentation is centered on the linear problem which provides good initial guesses for orbits of small amplitude, like the requirements for the mission Herschel/Planck of the European Space Agency (ESA), where part of the results of this paper were used for the mission design.

A kind of systems on the sphere, whose trajectories are similar to the Lissajous curves, are studied by means of one example. The symmetries are constructed following a unified and straightforward procedure for both the quantum and the classical versions of the model. In the quantum case it is stressed how the symmetries give the degeneracy of each energy level. In the classical case it is shown how the constants of motion supply the orbits, the motion and the frequencies in a natural way., Comment: 16 pages, 3 figures. Submitted