Finite Motion Research Articles

Anomalous Random Flights and Time-Fractional Run-and-Tumble Equations

Random flights (also called run-and-tumble walks or transport processes) represent finite velocity random motions changing direction at any Poissonian time. These models in d-dimension, can be studied giving a general formulation of the problem valid at any spatial dimension. The aim of this paper is to extend this general analysis to time-fractional processes arising from a non-local generalization of the kinetic equations. The probabilistic interpretation of the solution of the time-fractional equations leads to a time-changed version of the original transport processes. The obtained results provide a clear picture of the role played by the time-fractional derivatives in this kind of random motions. They display an anomalous behavior and are useful to describe several complex systems arising in statistical physics and biology. In particular, we focus on the one-dimensional random flight, called telegraph process, studying the time-fractional version of the classical telegraph equation and providing a suitable interpretation of its stochastic solutions.

Mobility of geometric constraint systems with extrusion symmetry

AbstractIf we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector $$\tau $$ τ , and finally join corresponding points of the two copies, then we obtain a framework with ‘extrusion’ symmetry in the direction of $$\tau $$ τ . This process may be repeated t times to obtain a framework whose underlying graph has $$\mathbb {Z}_2^t$$ Z 2 t as a subgroup of its automorphism group and which has ‘t-fold extrusion’ symmetry. Extruding a framework is a widely used technique in CAD for generating a 3D model from an initial 2D sketch, and hence it is important to understand the flexibility of extrusion-symmetric frameworks. Using group representation theory, we show that while t-fold extrusion symmetry is not a point-group symmetry, the rigidity matrix of a framework with t-fold extrusion symmetry can still be transformed into a block-decomposed form in the analogous way as for point-group symmetric frameworks. This allows us to establish Fowler-Guest-type character counts to analyse the mobility of such frameworks. We show that this entire theory also extends to the more general point-hyperplane frameworks with t-fold extrusion symmetry. Moreover, we show that under suitable regularity conditions the infinitesimal flexes we detect with our symmetry-adapted counts extend to finite (continuous) motions. Finally, we establish an algorithm that checks for finite motions via linearly displacing framework points along velocity vectors of infinitesimal motions.

Derivation of Miller’s rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator

Miller’s rule is an empirical relation between the nonlinear and linear optical coefficients that applies to a large class of materials but has only been rigorously derived for the classical Lorentz model with a weak anharmonic perturbation. In this work, we extend the proof and present a detailed derivation of Miller’s rule for an equivalent quantum-mechanical anharmonic oscillator. For this purpose, the classical concept of velocity-dependent damping inherent to the Lorentz model is replaced by an adiabatic switch-on of the external electric field, which allows a unified treatment of the classical and quantum-mechanical systems using identical potentials and fields. Although the dynamics of the resulting charge oscillations, and hence the induced polarizations, deviate due to the finite zero-point motion in the quantum-mechanical framework, we find that Miller’s rule is nevertheless identical in both cases up to terms of first order in the anharmonicity. With a view to practical applications, especially in the context of ab initio calculations for the optical response where adiabatically switched-on fields are widely assumed, we demonstrate that a correct treatment of finite broadening parameters is essential to avoid spurious errors that may falsely suggest a violation of Miller’s rule, and we illustrate this point by means of a numerical example.

Resonant phenomena in finite motions of test particles in oscillating dark matter configurations

Resonant phenomena in finite motions of test particles in oscillating dark matter configurations

The geometry of the Rindler black hole

Using Einstein's equations for gravity for a spherically symmetric field in a static state and the mass function method, a general form of the metric for Schwarzschild-type black holes is proposed. A qualitative analysis of Schwarzschild, Reissner–Nordström and Rindler geometries is performed. In the paper, it is found that Rindler black hole geometry is structurally inverse to Reissner–Nordström geometry, i.e. it has two T-regions and one R-region. In this regard Rindler geometry is like geometry of the Kottler black hole, which does not have flat asymptotics. Potential curves in the gravitational field of the Rindler black hole are constructed, which makes it possible to calculate the limits of finite motion for a test particle in this field. In addition, a qualitative comparison with the known Schwarzschild field metric is performed. Their similarity in behavior at small distances and absolutely different behavior at large scales are shown. Using the known accelerations of the Pioneer spacecraft, limits are numerically found for the T- and R-regions of their motion in the gravitational field of the Solar System.

Efficient Kinematic Calibration for Articulated Robot Based on Unit Dual Quaternion

Removing the parameter redundancy in the kinematic error model does improve the robustness of parameter identification. However, this does not help much on saving the calculation time on the error compensation process, which is closely related to the data structure used for representing rigid-body motion. Compared with the homogenous transformation matrix (HTM), the unit dual quaternion (UDQ) can describe the rigid-body motion in term of an array with eight entries, thus it is a conducive data structure for efficient kinematic computation. In this work, the exponential and logarithm mappings of a UDQ are defined explicitly, so that the relationship between finite and instantaneous motions is set up under the exponential coordinate. Thereafter, by defining the adjoint operator of UDQ to transform the twist under different frames, the linearized kinematic error models are built with local product-of-exponentials (POE) formula. This facilitates the upcoming parameter identification and error compensation processes. Specially, the counts of elementary operations are evaluated throughout the compensation processes, which shows the number of arithmetic operations are greatly reduced compared with using POE formula based on HTM. Additionally, the simple data structure in term of 1-D arrays saves the data dispatch time when performing adjoint operations. These make the developed model suitable for calibration of articulated robot with a long sequence of trajectories. Experiments on a 6-degree-of-freedom industrial robot validate the effectiveness of proposed approach.

“Clitic climbing” in Hittite

Abstract Cross-linguistically, clitic climbing occurs when clitics that belong syntactically and semantically to the subordinate clause (most commonly non-finite, rarely finite) appear in the main clause, i.e., they climb out of the subordinate clause into the main clause. In Hittite, prototypical clitic climbing is attested in two constructions: with non-finite predicates and finite restructuring verbs (Lyutikova & Sideltsev 2021b); and in serial constructions with the finite motion verbs pai- ‘go’ and uwa- ‘come’ co-occurring with another finite verb in the same clause (Koller 2013). In both of these cases, clitics climb out of complements of finite verbs. This paper explores yet another potentially relevant context for clitic climbing, a particular type of complex sentence that may be called ‘mismatch sentences’ (Sideltsev 2023). These involve three structurally distinct types of complex sentences which share one common property: they all have the same surface structure (1a) one word of the main clause (2) subordinate clause (1b) rest of the main clause. The enclitics of the subordinate clause are in (1a), so that they appear to climb out of the subordinate into the main clause. The enclitics of the main clause are consistently in the rest of the main clause (1b), never attached to the first word of the main clause (1a). Structurally, all these subordinate clauses adjoin to the main clause. This distribution of clitics is attested only if there is a one-word constituent in the main clause to the left of the subordinate clause. As movement out of adjoined clauses is held to be illicit in current linguistic theory, it is argued that, differently from prototypical clitic climbing, this is a purely post-syntactic reordering and does not involve any kind of syntactic movement of clitics out of the subordinate into the main clause: the structure one-word constituent of main clause—subordinate clause—main clause is always prosodically realized at the post-syntactic stage as subordinate clause—main clause.

Configuration Synthesis Method of Multi-Mode Parallel Mechanism Based on Variable Mobility Branch

Abstract In order to obtain more configurations of multi-mode parallel mechanism with the expected motion modes, a new configuration synthesis method of multi-mode parallel mechanism is proposed in this paper. The generation principle of parallel mechanism with multiple motion modes is analyzed. The relation between the motion of moving platform and the motion of branch is obtained. The finite motion and instantaneous motion of existing variable mobility branches are analyzed, and variable mobility branches are decomposed into variable mobility generators. By using variable mobility generators to construct variable mobility branches, a configuration synthesis method of multi-mode parallel mechanism based on variable mobility branch is proposed. Taking the multi-mode parallel mechanism with 2T1R and 2R1T motion modes as an example, a new class of multi-mode parallel mechanisms is obtained. The proposed method enriches the configuration synthesis theory of multi-mode parallel mechanism and the synthesis results enrich the configurations of multi-mode parallel mechanism.

Structural synthesis of the reconfigurable legged mobile lander for strong geometry-physical interaction with extraterrestrial environment

To design the engineering prototype of reconfigurable legged mobile lander (ReLML) for strong geometry-physical interaction with extraterrestrial environment, this paper proposes a tailored structural synthesis methodology unifying kinematic geometry and statics. Compared with our preceding works, three inbuilt operation modes are first integrated: adjusting mode is a totally new mode defined as three rotations (3R) with a strong geometrical adaptation to hostile terrain; landing mode is redefined as two rotations one translation (2R1T), inherits Chang'E lander to survive from strong impact interaction; roving mode turns into the other 2R1T motion for good motion/force interactivity. The interaction characteristics (motion, force, energy) between ReLML and environment are extracted to derive output finite motions and instantaneous wrenches of leg end-effector, following the principles of optimal force system equilibrium and work efficiency. The motion and force synergistic syntheses of metamorphic limb and joint are implemented to design and assemble mechanical generators. The ReLML can serve the alien base construction and adjustable launch bracket. The unified structural synthesis methodology of kinematic geometry and statics is universal and effective for designing parallel mechanisms and reconfigurable mechanisms, especially for strong physical interaction scenarios.

Task-Oriented Topology System Synthesis of Reconfigurable Legged Mobile Lander Integrating Active and Passive Metamorphoses

To explore hostile extraterrestrial landforms and construct an engineering prototype, this paper presents the task-oriented topology system synthesis of reconfigurable legged mobile lander (ReLML) with three operation modes from adjusting, landing, to roving. Compared with our preceding works, the adjusting mode with three rotations (3R) provides a totally novel exploration approach to geometrically matching and securely arriving at complex terrains dangerous to visit currently; the landing mode is redefined by two rotations one translation (2R1T), identical with the tried-and-tested Apollo and Chang’E landers to enhance survivability via reasonable touchdown buffering motion; roving mode also utilizes 2R1T motion for good motion and force properties. The reconfigurable mechanism theory is first brought into synthesizing legged mobile lander integrating active and passive metamorphoses, composed of two types of metamorphic joints and metamorphic execution and transmission mechanisms. To reveal metamorphic principles with multiple finite motions, the finite screw theory is developed to present the procedure from unified mathematical representation, modes and source phase derivations, metamorphic joint and limb design, to final structure assembly. To identify the prototype topology, the 3D optimal selection matrix method is proposed considering three operation modes, five evaluation criteria, and two topological subsystems. Finally, simulation verifies the whole task implementation process to ensure the reasonability of design.

Generalized Forms of an Overconstrained Sliding Mechanism Consisting of Two Congruent Tetrahedra

The motions of a bar structure consisting of two congruent tetrahedra are investigated, whose edges in their basic position are the face diagonals of a rectangular parallelepiped. The constraint of the motion is the following: the originally intersecting edges have to remain coplanar. All finite motions of our bar structure are determined. This generalizes our earlier work, where we did the same for the case when the rectangular parallelepiped was a cube. At the end of the paper we point out three further possibilities to generalize the question about the cube, and give for them examples of finite motions.

Type synthesis of parallel mechanism with 3T/3R motion patterns

As the competitor of 6-degree-of-freedom parallel mechanisms (PMs) in multi-tasks, parallel mechanisms with multi-motion patterns (PMMPs) have aroused considerable interest from researchers. This article investigates the type synthesis of parallel mechanisms with both three translational (3T) and three rotational (3R) motion patterns based on switch manner design. Having addressed the fundamentals of finite and instantaneous screws with regard to motion computations and differential mapping, the relationship between motion patterns is analyzed in the view of algebraic transformation operations. Then switch manners are discussed including both finite motion-driven and instantaneous singular motion-driven methods. An integrated expression is formulated to describe the whole moving process by defining transformation items. In this way, the type synthesis is carried out through four steps: switch manner design, integrated motion description, limb generation, and assembly condition solving. Numerous parallel mechanisms with 3T/3R motions are obtained and a typical application for space robots is presented. In this approach, different motion patterns are connected algebraically, and switch manners driven by both singular motion and finite motion are unified, which realize a unified and direct type synthesis process.

К динамике твердого стержня в пространстве Лобачевского

An absolutely solid infinitely thin rectilinear immaterial finite length rod helical motion problem is considered. Motion is heical relative to the inertia center. Located at rod ends point masses are identiall. The rod movement occurs in the constant negative curvature space (Lobachevsky space) under the gyroscopic forces system influence and a constant tracking force. The gyroscopic forces structure is set with using special conditions, which contain six characteristic constant parameters (gyroscopic coefficients). The characteristic motion properties are given. Components analytical dependense on the kinematic rod screw, rod orientation parameters is founded. Expressions of these parameters as time function, are obtained.

Quantification of left ventricular strain and torsion by joint analysis of 3D tagging and cine MR images.

Cardiovascular magnetic resonance (CMR) imaging is the gold standard for the non-invasive assessment of left-ventricular (LV) function. Prognostic value of deformation metrics extracted directly from regular SSFP CMR images has been shown by numerous studies in the clinical setting, but with some limitations to detect torsion of the myocardium. Tagged CMR introduces trackable features in the myocardium that allow for the assessment of local myocardial deformation, including torsion; it is, however, limited in the quantification of radial strain, which is a decisive metric for assessing the contractility of the heart. In order to improve SSFP-only and tagged-only approaches, we propose to combine the advantages of both image types by fusing global shape motion obtained from SSFP images with the local deformation obtained from tagged images. To this end, tracking is first performed on SSFP images, and subsequently, the resulting motion is utilized to mask and track tagged data. Our implementation is based on a recent finite element-based motion tracking tool with mechanical regularization. Joint SSFP and tagged images registration performance is assessed based on deformation metrics including LV strain and twist using human and in-house porcine datasets. Results show that joint analysis of SSFP and 3DTAG images provides better quantification of LV strain and twist as either data source alone.

A spectral inspection for turbulence amplification in oblique shock wave/turbulent boundary layer interaction

Turbulence amplification and the large-scale coherent structures in shock wave/turbulent boundary layer interaction flows have been studied at length in previous research, while the direct association between these two flow features is still lacking. In the present study, the transport equation of turbulent kinetic energy spectra is derived and utilized to analyse the scale-by-scale energy budget across the interaction zone, enabling us to reveal the association between the genesis of the large-scale motions and the turbulence amplification. For the presently considered flow with incipient shock-induced separation, we identified in turbulent kinetic energy spectra distribution that the most energetic motions are converted from the near-wall small-scale motions to large-scale motions consisting of velocity streaks and cross-stream circulations as they go through the interaction zone. The amplification of streamwise velocity fluctuation is triggered first, resulting in the emergence of large-scale velocity streaks, which is attributed to the adverse pressure gradient, as indicated by the spectra of the production term. The energy carried by large-scale velocity streaks is transferred to other velocity components by the pressure-strain term, producing large-scale cross-stream circulations. When large-scale motions are convected downstream, their energy is transferred via turbulent cascade to smaller scales and dissipated by viscosity. The spanwise uniform fluctuations, reminiscent of the unsteadiness of the separation bubble, are contributed primarily by the inter-scale energy transfer from the finite spanwise scale motions.

Multi-Flexible Body Dynamics Modeling and Experimental Study of the Patient Rehabilitation Transfer Device.

Objectives The patient rehabilitation transfer device is a typical personnel transfer equipment, which is mainly composed of outriggers, support arm, lifting arms, hooks, handrails, hydraulic cylinders, and other components. The existing research on the device is mainly focused on the configuration design and transfer mode, and the research on its dynamic characteristics during the transfer process has not been thoroughly discussed. Methodology. Based on the existing research, a portable hydraulic rehabilitation patient transfer device has been developed. Then the multi-rigid body dynamic and finite element flexible body models were established. Next, the dynamic characteristic difference between the two models of the device was studied. Results As shown in the results, the finite element multi-flexible body model has obvious flexible vibration in the lifting stage, and the amplitude reaches 16 mm in the motion startup stage due to the influence of rigid-flexible coupling. The tip acceleration of the flexible body model was also influenced by the vibration, and the maximum acceleration value reaches 0.06 m/s2. According to the test results, the maximum acceleration of the terminal reaches 0.05 m/s2, which is close to the finite-element multi-flexible model simulation results. The experimentally measured natural frequency of the device is 3.1 Hz, which is also close to 3.2 Hz calculated by the simulation. Because the flexible component in the flexible model is only the lift arm, the natural frequency is slightly larger than the experimental value. Conclusion According to the stress value of the finite element multi-body model motion process, the maximum stress appears at the moment when the motion reaches the top end, and the instantaneous stress reaches 206 Mpa, which is in line with the allowable stress range of the material design. The data obtained in this study will provide help for the follow-up clinical rehabilitation and intelligent device research.

Eddy saturation in a reduced two-level model of the atmosphere

Eddy saturation describes the nonlinear mechanism in geophysical flows whereby, when average conditions are considered, direct forcing of the zonal flow increases the eddy kinetic energy, while the energy associated with the zonal flow does not increase. Here, we present a minimal baroclinic model that exhibits complete eddy saturation. Starting from Phillips' classical quasi-geostrophic two-level model on the beta channel of the mid-latitudes, we derive a reduced order model comprising of six ordinary differential equations including parameterised eddies. This model features two physically realisable steady state solutions, one a purely zonal flow and one where, additionally, finite eddy motions are present. As the baroclinic forcing in the form of diabatic heating is increased, the zonal solution loses stability and the eddy solution becomes attracting. After this bifurcation, the zonal components of the solution are independent of the baroclinic forcing, and the excess of heat in the low latitudes is efficiently transported northwards by finite eddies, in the spirit of baroclinic adjustment.

The Mathematical Model of Curve-Face Gear and Time-Varying Meshing Characteristics of Compound Transmission

The curve-face gear pair is a new type of gear pair with variable transmission ratio for spatial finite helical motion. In this paper, mathematical models of a new developed curve-face gear were simplified and obtained directly by the standard shaper. The subsequent studies on the curve-face gear compound transmission characteristics were further analyzed by the combinations of the principle of space gearing and screw theory. Firstly, the conjugate tooth surface geometry as well the point contact traces of curve-face gears were derived. Secondly, the geometric relationships between gear pair and the corresponding meshing characteristics were evaluated by several basic geometric elements, including instantaneous screw, axodes, striction curve, and conjugate pitch surface. Based on that analysis, it was found that the tooth contact normal was reciprocal to the instantaneous twist, which demonstrated that the conjugate motion with the desired transmission ratio could be realized in current curve-face gear compound transmission. Moreover, the time-varying slip characteristics of the curve-face gear pair were also revealed, that is, rolling and sliding action coexist at all contact on the tooth surfaces. In brief, this work provided the theoretical basis for following researches on machining curve-face gear with standard shaper.

Adjustable positive and negative hygrothermal expansion metamaterial inspired by the Maltese cross.

A metamaterial that can manifest both positive and negative coefficients of moisture and thermal expansion is presented herein, based on inspiration from the Maltese cross. Each unit of the metamaterial consists of a pair of equal-armed crosses pin-joined at their junctions to permit rotation, but elastically restrained by a bimaterial spiral spring, and four pairs of hinge rods to translate the relative rotational motion of the pair of equal-armed crosses into translational motion of the connecting rods. The effective coefficients of moisture and thermal expansion models were developed for small and large changes in the hygrothermal conditions using infinitesimal (approximate) and finite (exact) motion analyses, respectively, with the former giving constant effective coefficients with respect to environmental changes. Results indicate that the approximate method underestimates the magnitude of both the effective expansion coefficients under cooling and drying but overestimates magnitudes of both coefficients during heating and moistening, and that the change in both expansion coefficients is more drastic during cooling and drying than during heating and moistening. In addition to providing another micro-lattice geometry for effecting expansion coefficients of either signs, this metamaterial exhibits auxetic property.

Simulation of near-fault ground strains and rotations from actual strike-slip earthquakes: case studies of the 2004Mw 6.0 Parkfield, the 1979Mw 6.5 Imperial Valley and the 1999Mw 7.5 Izmit earthquakes

SUMMARYPrevious studies have demonstrated that finite-fault simulations of actual or hypothetical earthquakes using deterministic, physics-based simulation techniques constitute an effective tool for characterizing near-fault ground strains and rotations in the low-frequency range. The characteristics of these motions are further investigated in this study by performing forward ground-motion simulations of three well-documented strike-slip earthquakes (i.e. 2004 Mw 6.0 Parkfield, 1979 Mw 6.5 Imperial Valley and 1999 Mw 7.5 Izmit) using models of the seismic source and crustal structure available in the literature. Time histories of ground strains and rotations are numerically generated at near-fault stations and at a dense grid of observation points extending over the causative fault. This is achieved by finite differencing translational motions simulated at very closely spaced stations using a kinematic modelling approach. The simulation results show that the three strike-slip earthquakes produce large-amplitude pulse-like shear strain and torsion in the forward direction of rupture propagation. The time histories of specific components of displacement gradient, strain and rotation at near-fault stations can be estimated from those of ground velocities using a phase velocity, whereas peak ground torsions in the near-fault region can be reasonably estimated from peak horizontal ground velocities using a scaling factor. However, both the phase velocity and the scaling factor exhibit significant variability in the near-fault region of the considered earthquakes. The concept of isochrones is also utilized to associate fault rupture characteristics with near-fault ground strains and rotations. The results indicate that the seismic energy radiated from the high-isochrone-velocity region of the fault—which encompasses areas of large slip locally driven by high stress drop—arrives at a near-fault station in a short time interval that coincides with the time window of the large-amplitude pulse-like shear strain and torsion.