The statistical distribution of the length of a rubber molecule

1948 ◽  
Vol 44 (3) ◽  
pp. 342-344 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Normal Distribution ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Statistical Distribution ◽  
Equal Length ◽  
Fixed Angle ◽  
A Chain ◽  
Image Position ◽  
Three Degrees Of Freedom

A rubber molecule containing n + 1 carbon atoms may be represented by a chain of n links of equal length such that successive links are at a fixed angle to each other but are otherwise at random. The statistical distribution of the length of the molecule, that is, the distance between the first and last carbon atoms, has been considered by various authors (Treloar (1) gives references). In particular, if the first atom is kept fixed at the origin of a system of coordinates and the chain is otherwise at random, it has been conjectured that the distribution of the (n + 1)th atom will tend, as n increases, towards a three-dimensional normal distribution of the formwhere σ depends on n. Thus r2 (= x2 + y2 + z2) will be approximately distributed as σ2χ2 with three degrees of freedom.

scholarly journals Carter's constant and superintegrability

2018 ◽  
Vol 27 (07) ◽  
pp. 1850066
Author(s):  
Payel Mukhopadhyay ◽  
K. Rajesh Nayak
Keyword(s):  
Dynamical System ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Integrals Of Motion ◽  
Superintegrable System ◽  
Simple Observation ◽  
Superintegrable Systems ◽  
Axisymmetric Spacetime ◽  
Three Degrees Of Freedom

Carter's constant is a nontrivial conserved quantity of motion of a particle moving in stationary axisymmetric spacetime. In the version of the theorem originally given by Carter, due to the presence of two Killing vectors, the system effectively has two degrees of freedom. We propose an extension to the first version of Carter's theorem to a system having three degrees of freedom to find two functionally independent Carter-like integrals of motion. We further generalize the theorem to a dynamical system with [Formula: see text] degrees of freedom. We further study the implications of Carter's constant to superintegrability and present a different approach to probe a superintegrable system. Our formalism gives another viewpoint to a superintegrable system using the simple observation of separable Hamiltonian according to Carter's criteria. We then give some examples by constructing some two-dimensional superintegrable systems based on this idea and also show that all three-dimensional simple classical superintegrable potentials are also Carter separable.

Thermogasodynamic Analysis of the Segmented Annular Seal

2020 ◽  
Vol 143 (7) ◽  
Author(s):  
Samia Dahite ◽  
Mihai Arghir
Keyword(s):  
Parametric Study ◽  
Thermal Effects ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Leading Edge ◽  
Test Case ◽  
Isothermal Model ◽  
Pocket Depth ◽  
Annular Seal ◽  
Three Degrees Of Freedom

Abstract The present work deals with the thermogasodynamic analysis of the segmented annular seal provided with Rayleigh pockets. The paper is a continuation of the work presented Arghir, M., and Mariot, A. (2017, “Theoretical Analysis of the Static Characteristics of the Carbon Segmented Seal,” ASME J. Tribol., 139(6), p. 062202.) where an isothermal model of the segmented annular seal was first presented. Each segment had three degrees-of-freedom, and its static position was obtained by solving the nonlinear equations of equilibrium. Thermal effects are now introduced by considering a simplified form of the energy equation in the thin gas film coupled with the three dimensional heat transfer in a segment of the seal and in the rotor. An efficient numerical algorithm is developed. A parametric study was performed for a segmented annular seal with pockets taken from the literature and operating with air. First, a test case proved the necessity of considering three degrees-of-freedom for the segment and not only its radial displacement. The parametric study was then performed for two different pocket depths, two pressure differences, and different rotation speeds. The results showed a non-uniform heating with larger temperatures at the leading edge of the segment where the minimal film thickness occurs. Heating is proportional to the pocket depth that lowers the lift force of the segment and to the pressure difference that closes the seal.

Near-Integrability and Recurrence in FPU-Cells

2016 ◽  
Vol 26 (14) ◽  
pp. 1650230 ◽  
Author(s):  
Ferdinand Verhulst
Keyword(s):  
Hamiltonian Systems ◽  
Building Block ◽  
Degrees Of Freedom ◽  
Stable Equilibrium ◽  
Chain Recurrence ◽  
Low Dimension ◽  
A Cell ◽  
Linearized System ◽  
A Chain ◽  
Three Degrees Of Freedom

In a neighborhood of stable equilibrium, we consider the dynamics for at least three degrees-of-freedom (dof) Hamiltonian systems (2 dof systems are not ergodic in this case). A complication is that the recurrence properties depend strongly on the resonances of the corresponding linearized system and on quasi-trapping. In contrast to the classical FPU-chain, the inhomogeneous FPU-chain shows nearly all the principal resonances. Using this fact, we construct a periodic FPU-chain of low dimension, called a FPU-cell. Such a cell can be used as a building block for a chain of FPU-cells, called a cell-chain. Recurrence phenomena depend strongly on the physical assumptions producing specific Hamiltonians; we demonstrate this for the [Formula: see text] resonance, both general and for the FPU case; this resonance shows dynamics on different timescales. In addition we will study the relations and recurrence differences between several FPU-cells and a few cell-chains in the case of the classical near-integrable FPU-cell and of chaotic cells in [Formula: see text] resonance.

scholarly journals Contact Force Control of Three-Dimensional Flexible Manipulator

2020 ◽  
Author(s):  
Minoru Sasaki ◽  
Shunta Ito ◽  
Daiki Maeno ◽  
Waweru Njeri ◽  
Muguro Josephh ◽  
...  
Keyword(s):  
Contact Force ◽  
Force Control ◽  
Degrees Of Freedom ◽  
Force Feedback ◽  
Three Dimensional ◽  
Flexible Manipulator ◽  
Force Tracking ◽  
Feedback Method ◽  
Three Degrees Of Freedom ◽  
Contact Force Control

This paper proposes a contact force controller for a constrained flexible manipulator in three-dimensional motion. This controller used the conversion formula obtained empirically and experimental results showed the effectiveness of the proposed contact force controller. First, the manipulator was operated with the tip of the second link restrained, then, time response of the root strain, joint angles and contact force were used to derive the relational between the three quantities. The effectiveness of the relational expression was verified by conducting a target contact force tracking experiment by inputting the angle from the relational expression. The contact force control using the strain feedback method was proposed with the strain amount estimated from the target contact force as the target value, and its effectiveness was verified by experiments. From the results obtained, controller using the strain feedback method was designed for the purpose of controlling the contact force at the tip of a flexible manipulator with two links and three degrees of freedom that performs three-dimensional spatial motion, and its effectiveness was shown by comparison with the contact force feedback method.

A First Example: The “Particle in a Box“ and Quantized Translational Motion

2020 ◽  
pp. 59-96
Author(s):  
Jochen Autschbach
Keyword(s):  
Degrees Of Freedom ◽  
Separation Of Variables ◽  
Translational Motion ◽  
Three Dimensional ◽  
Potential Wells ◽  
Quantum Tunnelling ◽  
Atomic Force ◽  
Quantum Behavior ◽  
The One ◽  
Three Degrees Of Freedom

The simple ‘particle in a box’ (Piab) is introduced in this chapter so that the reader can get familiar with applying the quantum recipe and atomic units. The PiaB is introduced in its one, two, and three dimensional variants, which demonstrates the use of the separation of variables technique as a strategy to solve the Schrodinger equation for a particle with two or three degrees of freedom. It is shown that the confinement of the particle causes the energy to be quantized. The one-dimensional PiaB is then applied to treat the electronic spectra of cyanine dyes and their absorption colors. The chapter then introduces more general setups with finite potential wells, in order to introduce the phenomenon of quantum tunnelling and to discuss more generally with the unintuitive ‘quantum behavior’ of particles such as electrons. Scanning tunnelling and atomic force microscopes are also discussed briefly.

Static and Vibration Analyses of a Three-Dimensional Snake-Like Micropositioning Stage

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Nicola Scuor ◽  
Paolo Gallina ◽  
Marco Giovagnoni
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Dynamic Models ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Dynamic Tests ◽  
Lumped Mass ◽  
Simulation Data ◽  
End Effector ◽  
Three Degrees Of Freedom

This paper presets three degrees of freedom (DOF) piezoelectric micropositioning stage. The stage is composed of a stack of piezodisk bender actuators actuated in such a way to prevent the end-effector from rotating; this way the end-effector can only translate along the x, y, and z axes. Thanks to its snake-like configuration, the system is capable of large displacements (of the order of 50 μm) with low driving voltages (of the order of 100 V). Several lumped-mass static and dynamic models of the device have been implemented. Static experimental results, which are in agreement with simulation data, confirmed the performances of the device. A dynamic model showed the natural frequencies of the mechanism. Also dynamic tests have been conducted in order to validate the dynamic model.

The Qualitative Synthesis of Parallel Manipulators

2004 ◽  
Vol 126 (4) ◽  
pp. 617-624 ◽  
Author(s):  
Jorge Angeles
Keyword(s):  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Qualitative Reasoning ◽  
Rigid Bodies ◽  
Three Dimensions ◽  
Qualitative Synthesis ◽  
Motion Representation ◽  
Six Degrees Of Freedom ◽  
Three Degrees Of Freedom

As shown in this paper, when designing parallel manipulators for tasks involving less than six degrees of freedom, the topology can be laid out by resorting to qualitative reasoning. More specifically, the paper focuses on cases whereby the manipulation tasks pertain to displacements with the algebraic structure of a group. Besides the well-known planar and spherical displacements, this is the case of displacements involving: rotation about a given axis and translation in the direction of the same axis (cylindrical subgroup); translation in two and three dimensions (two- and three-dimensional translation subgroups); three independent translations and rotation about an axis of fixed direction, what is known as the Scho¨nflies subgroup; and similar to the Scho¨nflies subgroup, but with the rotation and the translation in the direction of the axis of rotation replaced by a screw displacement. For completeness, the fundamental concepts of motion representation and groups of displacements, as pertaining to rigid bodies, are first recalled. Finally, the concept of Π-joint, introduced elsewhere, is generalized to two and three degrees of freedom, thereby ending up with the Π2-and the Π3-joints, respectively.

scholarly journals Design and Evaluation of a Robotic Device for Automated Tail Vein Cannulations in Rodent Models

2017 ◽  
Vol 11 (4) ◽  
Author(s):  
Alex Fromholtz ◽  
Max L. Balter ◽  
Alvin I. Chen ◽  
Josh M. Leipheimer ◽  
Anil Shrirao ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Near Infrared ◽  
Three Dimensional ◽  
Needle Insertion ◽  
Rodent Models ◽  
Tail Vein ◽  
Work Volume ◽  
Stereo Cameras ◽  
And Training ◽  
Three Degrees Of Freedom

Preclinical testing in rodent models is a ubiquitous part of modern biomedical research and commonly involves accessing the venous bloodstream for blood sampling and drug delivery. Manual tail vein cannulation is a time-consuming process and requires significant skill and training, particularly since improperly inserted needles can affect the experimental results and study outcomes. In this paper, we present a miniaturized, robotic medical device for automated, image-guided tail vein cannulations in rodent models. The device is composed of an actuated three degrees-of-freedom (DOFs) needle manipulator, three-dimensional (3D) near-infrared (NIR) stereo cameras, and an animal holding platform. Evaluating the system through a series of workspace simulations and free-space positioning tests, the device exhibited a sufficient work volume for the needle insertion task and submillimeter accuracy over the calibration targets. The results indicate that the device is capable of cannulating tail veins in rodent models as small as 0.3 mm in diameter, the smallest diameter vein required to target.

A Spherical DC Servo Motor With Three Degrees of Freedom

1989 ◽  
Vol 111 (3) ◽  
pp. 398-402 ◽  
Author(s):  
K. Kaneko ◽  
I. Yamada ◽  
K. Itao
Keyword(s):  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Matrix Elements ◽  
Servo Motor ◽  
Constant Matrix ◽  
Spherical Motor ◽  
The Matrix ◽  
Three Degrees Of Freedom

A spherical DC servo motor with three degrees of freedom is proposed. First, the process of generating three-dimensional torque is analyzed to obtain the torque constant matrix. The matrix elements are shown to vary with rotor inclination, and winding currents are shown to interfere with each other. Then, the dynamics of the spherical motor are investigated theoretically and experimentally, considering torque interference, gyro moment and gravity. Finally, the trajectory of the prototype motor is shown in order to clarify its abilities. This new spherical motor is expected to produce a smaller, a lighter mechanism, since no gears or linkages are needed.

scholarly journals A gravity-based three-dimensional compass in the mouse brain

2019 ◽  
Author(s):  
Dora E Angelaki ◽  
J Ng ◽  
AM Abrego ◽  
HX Cham ◽  
JD Dickman ◽  
...  
Keyword(s):  
Horizontal Plane ◽  
Degrees Of Freedom ◽  
Mammalian Species ◽  
Three Dimensional ◽  
Retrosplenial Cortex ◽  
Orientation Tuning ◽  
Head Direction ◽  
Head Direction Cells ◽  
Dimensional Gravity ◽  
Three Degrees Of Freedom

SummaryHead direction cells in the mammalian limbic system are thought to function as an allocentric neuronal compass. Although traditional views hold that the compass of ground-dwelling species is planar, we show that head-direction cells in the rodent thalamus, retrosplenial cortex and cingulum fiber bundle are tuned to conjunctive combinations of azimuth, pitch or roll, similarly to presubicular cells in flying bats. Pitch and roll orientation tuning is ubiquitous, anchored to gravity, and independent of visual landmarks. When head tilts, azimuth tuning is affixed to the head-horizontal plane, but also uses gravity to remain anchored to the terrestrial allocentric world. These findings suggest that gravity defines all three degrees of freedom of the allocentric orientation compass, and only the azimuth component can flexibly remap to local cues in different environments. Collectively, these results demonstrate that a three-dimensional, gravity-based, neural compass is likely a ubiquitous property of mammalian species, including ground-dwelling animals.

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