Curvature Theory on Contact and Transfer Characteristics of Enveloping Curves

2018 ◽  
Author(s):  
Kwun-Lon Ting ◽  
Cody Leeheng Chan
Keyword(s):  
Relative Motion ◽  
Contact Point ◽  
Rigid Bodies ◽  
Body Motion ◽  
Transfer Characteristics ◽  
Moving Body ◽  
Curvature Theory ◽  
The Veil ◽  
Motion Characteristics ◽  
Definition Of

In differential geometry, a curve is characterized by the curvature properties and so is a point trajectory in curvature theory. However, due to the rolling and sliding between contact curves, the characterization of enveloping curves embedded on rigid bodies in relative motion is not complete without the transfer (or shifting) characteristics of the contact point. This paper presents the new perspectives and the first comprehensive theory on not only the curvature characteristics but also the transfer characteristics between enveloping curves embedded on rigid bodies. The paper contains three parts. In the first part, a point traces a curve on the moving body and consequently traces a curve on the fixed body. Both generated curves form a pair of enveloping curves. This part establishes the foundation of the paper. Because each enveloping curve is treated as a point trajectory. One may examine all aspects of the enveloping process. Essentially this unmasks the veil that has hindered further understanding and observation of the enveloping behavior beyond the fundamental curvature. It represents a significant advancement on envelope theory. In the second part, the moving point is the instant center, which traces the moving centrode on the moving body and the fixed centrode on the fixed body. It characterizes the rolling between centrodes and the transfer characteristics of the instant center on each centrode. It not only offers a simple way to treat the instant center transfer (shifting) velocity but also successfully extends it to any order of motion. The third part is about the rolling and sliding of between enveloping curves embedded on rigid bodies in relative motion. It addresses the transfer characteristics of the contact on each of the contact curves for the first time. The transfer characteristics are functions of the rigid body motion characteristics. This part offers the vital kinematic aspect of enveloping curves distinctly different from the conventional curvature theory that addresses an individual curve. The proposed enveloping curvature theory offers an important model to account for all aspects of the contact and removes the veil that blurs the contact behavior caused by the traditional envelope definition of Fλxy=∂F∂λλxy=0. This is a kinematic solution for envelopes. The proposed theory is illustrated with an example of two rolling cylinders.

Curvature Theory on Contact and Transfer Characteristics of Enveloping Curves

2019 ◽  
Vol 12 (1) ◽  
Author(s):  
Cody Leeheng Chan ◽  
Kwun-Lon Ting
Keyword(s):  
Transfer Rate ◽  
Contact Point ◽  
Kinematic Model ◽  
Rigid Bodies ◽  
Transfer Velocity ◽  
The Third ◽  
Transfer Characteristics ◽  
Curvature Theory ◽  
Transfer Properties

Abstract In differential geometry, a curve is characterized by the curvature properties and so is a point trajectory in curvature theory. However, due to the rolling and sliding between contact curves, the characterization of enveloping curves embedded on rigid bodies in relative motion is not complete without the transfer rate of the contact point. This paper presents the new perspectives on not only the curvature characteristics but also the transfer characteristics between enveloping curves embedded on rigid bodies. The paper contains three parts. First, a pair of enveloping curves can be described by a point trajectory which traces two curves simultaneously. Second, the paper treated instant center as a tracing point, which traces the moving centrode and the fixed centrode on each of the bodies. This characterizes the rolling between centrodes and the transfer rate of the instant center on each of centrodes. The treatment can extend the instant center transfer velocity to any higher order. The third part is about the rolling and sliding between enveloping curves embedded on rigid bodies motion. A kinematic model to describe the transfer rate of contact point based on the motion of the instant center is proposed. The detailed curvature and transfer properties of line- and circle-envelopes are presented. Due to the simple treatment, this might be the first paper that successfully achieved the curvature properties of circle-envelopes. The method offers the vital kinematic aspect of enveloping curves distinctly different from the conventional curvature theory and enveloping theory.

Contact Kinematics between Three-Dimensional Rigid Bodies with General Surface Parameterization

2020 ◽  
pp. 1-28
Author(s):  
Mubang Xiao ◽  
Ye Ding
Keyword(s):  
Reference Frame ◽  
Relative Motion ◽  
Screw Theory ◽  
Dimensional Space ◽  
Contact Point ◽  
Three Dimensional ◽  
Rigid Bodies ◽  
Frame Field ◽  
Contact Kinematics ◽  
Acceleration Analysis

Abstract This paper provides an improvement of the classic Montana's contact kinematics equations considering non-orthogonal object parameterizations. In Montana's model, the reference frame used to define the relative motion between two rigid bodies in three-dimensional space is chosen as the Gauss frame, assuming there is an orthogonal coordinate system on the object surface. To achieve global orthogonal parameterizations on arbitrarily shaped object surfaces, we define the relative motion based on the reference frame field, which is the orthogonalization of the surface natural basis at every contact point. The first- and second-order contact kinematics, including the velocity and acceleration analysis of the relative rolling, sliding, and spinning motion, are reformulated based on the reference frame field and the screw theory. We use two simulation examples to illustrate the proposed method. The examples are based on simple non-orthogonal surface parameterizations, instead of seeking for global orthogonal parameterizations on the surfaces.

Music and Motion——How Music-Related Ancillary Body Movements Contribute to the Experience of Music

2009 ◽  
Vol 26 (4) ◽  
pp. 335-353 ◽  
Author(s):  
Manfred Nusseck ◽  
Marcelo M. Wanderley
Keyword(s):  
Relative Motion ◽  
Body Motion ◽  
Whole Body ◽  
Body Parts ◽  
Body Movements ◽  
Musical Performances ◽  
Motion Characteristics

EXPRESSIVE PERFORMER MOVEMENTS IN MUSICAL performances represent implied levels of communication and can contain certain characteristics and meanings of embodied human expressivity. This study investigated the contribution of ancillary body movements on the perception of musical performances. Using kinematic displays of four clarinetists, perceptual experiments were conducted in which participants were asked to rate specific music-related dimensions of the performance and the performer. Additionally, motions of particular body parts, such as movements of the arms and torso, as well as motion amplitudes of the whole body were manipulated in the kinematic display. It was found that manipulations of arm and torso movements have fewer effects on the observers' ratings of the musicians than manipulations concerning the movement of the whole body. The results suggest that the multimodal experience of musicians is less dependent on the players' particular body motion behaviors than it is on the players' overall relative motion characteristics.

Generalized Vector Derivatives for Systems With Multiple Relative Motion

1968 ◽  
Vol 35 (1) ◽  
pp. 20-24 ◽  
Author(s):  
T. A. Sherby ◽  
J. F. Chmielewski
Keyword(s):  
Reference Frame ◽  
Relative Motion ◽  
Reference Frames ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Main Body ◽  
Angular Velocity Vector ◽  
Relative Coordinates ◽  
Moving Body ◽  
Classical Vector

For the analysis of relative motion, classical vector mathematics is limited to the use of one moving reference frame when taking vector derivatives. However, many dynamical systems consist of a number of rigid bodies in motion relative to one another. The classical procedure requires the specification of the position of each body relative to a single “main body.” The use of relative coordinates allows a natural specification of the position of one moving body relative to another moving body in network fashion. To use relative coordinates in dynamic and kinematic analyses, it is necessary to use relative vector derivatives involving more than one moving reference frame. This paper presents general expressions for the kth-order derivative of a relative position and angular velocity vector measured in any moving reference frame in a system of m reference frames with multiple relative motion. These expressions are used to develop a procedure which generates the differential equations of motion for the system by routine substitution of the relative coordinates and their scalar derivatives. This procedure offers promise as an algorithm for machine generation of the system equations and eliminates the possibility of neglecting subtle accelerations due to relative motion. The use of the procedure is demonstrated by generating the equations of motion of an offset unsymmetrical gyroscope.

scholarly journals Modeling and Prediction of Rigid Body Motion with Planar Non-Convex Contact

2021 ◽  
pp. 1-17
Author(s):  
Jiayin Xie ◽  
Nilanjan Chakraborty
Keyword(s):  
Rigid Body ◽  
Dynamic Simulation ◽  
Contact Region ◽  
Contact Point ◽  
Single Point ◽  
Point Contact ◽  
Rigid Bodies ◽  
Body Motion ◽  
Contact Patch ◽  
Connected Set

Abstract We present a principled method for motion prediction via dynamic simulation for rigid bodies in intermittent contact with each other where the contact region is a planar non-convex contact patch. Such methods are useful in planning and control for robotic manipulation. The planar non-convex contact patch can either be a topologically connected set or disconnected set. Most work in rigid body dynamic simulation assume that the contact between objects is a point contact, which may not be valid in many applications. In this paper, by using the convex hull of the contact patch, we build on our recent work on simulating rigid bodies with convex contact patches for simulating motion of objects with planar non-convex contact patches. We formulate a discrete-time mixed complementarity problem where we solve the contact detection and integration of the equations of motion simultaneously. We solve for the equivalent contact point (ECP) and contact impulse of each contact patch simultaneously along with the state, i.e., configuration and velocity of the objects. We prove that although we are representing a patch contact by an equivalent point, our model for enforcing non-penetration constraints ensure that there is no artificial penetration between the contacting rigid bodies. We provide empirical evidence to show that our method can seamlessly capture transition among different contact modes like patch contact, multiple or single point contact.

The Curvature Theory of Line Trajectories in Spatial Kinematics

1981 ◽  
Vol 103 (4) ◽  
pp. 718-724 ◽  
Author(s):  
J. M. McCarthy ◽  
B. Roth
Keyword(s):  
Planar Motion ◽  
Spatial Motion ◽  
Third Order ◽  
Local Shape ◽  
The Third ◽  
Moving Body ◽  
Physical Representation ◽  
Curvature Theory ◽  
Spatial Kinematics ◽  
Differential Properties

This paper develops the differential properties of ruled surfaces in a form which is applicable to spatial kinematics. Derivations are presented for the three curvature parameters which define the local shape of a ruled surface. Related parameters are also developed which allow a physical representation of this shape as generated by a cylindric-cylindric crank. These curvature parameters are then used to define all the lines in the moving body which instantaneously generate speciality shaped trajectories. Such lines may be used in the synthesis of spatial motions in the same way that the points on the inflection circle and cubic of stationary curvature are used to synthesize planar motion. As an example of this application several special sets of lines are defined: the locus of all lines which for a general spatial motion instantaneously generate helicoids to the second order and the locus of lines generating right hyperboloids to the third order.

Synthesis of Planar, Compliant Four-Bar Mechanisms for Compliant-Segment Motion Generation

1999 ◽  
Vol 123 (4) ◽  
pp. 535-541 ◽  
Author(s):  
L. Saggere ◽  
S. Kota
Keyword(s):  
Compliant Mechanisms ◽  
Shape Change ◽  
Synthesis Method ◽  
Elastic Equilibrium ◽  
Body Motion ◽  
Motion Generation ◽  
Generation Task ◽  
Potential Applications ◽  
Flexible Segment ◽  
Definition Of

Compliant four-bar mechanisms treated in previous works consisted of at least one rigid moving link, and such mechanisms synthesized for motion generation tasks have always comprised a rigid coupler link, bearing with the conventional definition of motion generation for rigid-link mechanisms. This paper introduces a new task called compliant-segment motion generation where the coupler is a flexible segment and requires a prescribed shape change along with a rigid-body motion. The paper presents a systematic procedure for synthesis of single-loop compliant mechanisms with no moving rigid-links for compliant-segment motion generation task. Such compliant mechanisms have potential applications in adaptive structures. The synthesis method presented involves an atypical inverse elastica problem that is not reported in the literature. This inverse problem is solved by extending the loop-closure equation used in the synthesis of rigid-links to the flexible segments, and then combining it with elastic equilibrium equation in an optimization scheme. The method is illustrated by a numerical example.

Unraveling Paradoxical Theories for Rigid Body Collisions

1991 ◽  
Vol 58 (4) ◽  
pp. 1049-1055 ◽  
Author(s):  
W. J. Stronge
Keyword(s):  
Frictional Force ◽  
Contact Point ◽  
Tangential Velocity ◽  
Velocity Slip ◽  
Rigid Bodies ◽  
Contact Points ◽  
Before And After ◽  
Limited Applicability ◽  
Work Done ◽  
Impulsive Reaction

A collision between two rigid bodies has a normal impulsive reaction at the contact point (CP). If the bodies are slightly rough and the contact points have a relative tangential velocity (slip), there is also a frictional force that opposes slip. Small initial slip can halt before contact terminates; when slip halts the frictional force changes and the collision process is separated into periods before and after halting. An energetically consistent theory for collisions with slip that halts is based on the work done by normal (nonfrictional) forces during restitution and compression phases. This theory clearly separates dissipation due to frictional forces from that due to internal irreversible deformation. With this theory, both normal and tangential components of the impulsive reaction always dissipate energy during collisions. In contrast, Newton’s impact law results in calculations of paradoxical increases in energy for collisions where slip reverses. This law relates normal components of relative velocity for the CP at separation and incidence by a constant (the coefficient of restitution e). Newton’s impact law is a kinematic definition for e that generally depends on the slip process and friction; consequently it has limited applicability.

A High Invariance Motion Representation for Skeleton-Based Action Recognition

2016 ◽  
Vol 30 (08) ◽  
pp. 1650018 ◽  
Author(s):  
Songrui Guo ◽  
Huawei Pan ◽  
Guanghua Tan ◽  
Lin Chen ◽  
Chunming Gao
Keyword(s):  
Action Recognition ◽  
Relative Motion ◽  
Local Coordinate System ◽  
Research Work ◽  
Human Action Recognition ◽  
Human Action ◽  
Body Motion ◽  
Motion Representation ◽  
Representation Method ◽  
Relative Geometry

Human action recognition is very important and significant research work in numerous fields of science, for example, human–computer interaction, computer vision and crime analysis. In recent years, relative geometry features have been widely applied to the description of relative relation of body motion. It brings many benefits to action recognition such as clear description, abundant features etc. But the obvious disadvantage is that the extracted features severely rely on the local coordinate system. It is difficult to find a bijection between relative geometry and skeleton motion. To overcome this problem, many previous methods use relative rotation and translation between all skeleton pairs to increase robustness. In this paper we present a new motion representation method. It establishes a motion model based on the relative geometry with the aid of special orthogonal group SO(3). At the same time, we proved that this motion representation method can establish a bijection between relative geometry and motion of skeleton pairs. After the motion representation method in this paper is used, the computation cost of action recognition reduces from the two-way relative motion (motion from A to B and B to A) to one-way relative motion (motion from A to B or B to A) between any skeleton pair, namely, permutation problem [Formula: see text] is simplified into combinatorics problem [Formula: see text]. Finally, the experimental results of the three motion datasets are all superior to present skeleton-based action recognition methods.

Offloading Operability of Small Scale AG FLNG With Side-by-Side Moored Small Scale LNG Carrier in Offshore West Africa

2017 ◽  
Author(s):  
Mun-sung Kim ◽  
Eric Morilhat ◽  
X. C. Nguyen ◽  
Bo-hee Kim ◽  
Jung-moon Jang ◽  
...  
Keyword(s):  
West Africa ◽  
Relative Motion ◽  
Structural Integrity ◽  
Effective Means ◽  
Production Capacity ◽  
Cost Effective ◽  
Body Motion ◽  
Small Scale ◽  
Associated Gas ◽  
High Level

This study describes one of the technical solutions for Small Scale FLNG (SSFLNG)[1] development specifically designed to monetize Associated Gas (AG) of producing oil fields located within convenient distance of an existing LNG Plant or Port with LNG storage facility. Limited production capacity combined with short range small scale LNG carriers (SSLNGC), provide a cost effective means for LNG production. Ship to ship off-loading operation by loading arm has been considered in AG SSFLNG. Produced LNG is to be off-loaded from the SSFLNG to side-by-side moored SSLNGC. Relative motion and dynamic load acting on loading arm system in side-by-side mooring arrangement is one of key factors to estimate the offloading operability of the AG SSFLNG. In this paper, a numerical two-body motion analysis for the side-by-side moored SSFLNG in frequency- and time-domain is carried out. Also, the basic engineering work is carried out for the marine loading arms (MLA). Since the MLA reacts approximately as a linear system, it is calculated by a full spectral RAO analysis for each of the worst load cases issued from the spectral ranking. All loads and stresses inside the MLA are verified in accordance with EN1474-1[2] for the situations identified in the previous step. A high level fatigue analysis focused on the cryogenic swivel joints is carried out. Based on the numerical calculation for relative motion in side-by-side moored FLNG, we have been performed structural assessment for MLA in several environment conditions. The structural integrity of both MLA and the LNGC manifold are validated during offloading for Offshore West Africa.

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