Locus of the Axis of Vibration for a Vibrating System With Variable Location of a Mass Center

2000 ◽  
Author(s):  
Byung Ju Dan ◽  
Yong Je Choi
Keyword(s):  
Vibration Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Analytical Form ◽  
Point Of View ◽  
Information Storage ◽  
Mode Shapes ◽  
Storage Device ◽  
Geometrical Approach ◽  
Mass Center

Abstract A typical approach to a linear vibration analysis of an elastically supported single rigid body is to rearrange a dynamic model into a corresponding eigenvalue problem. From the geometrical point of view, the eigenvectors in the planar vibration analysis can be interpreted as pure rotations about the vibration center or pure translations. In a three dimensional space, they represent repetitive twisting motions about the axes of vibrations. By taking a geometrical approach to the vibration analysis, the vibration mode shapes may be better understood. In this paper, the influence of variable location of a mass center on the locations of the axes of vibrations and the natural frequencies are investigated by means of the locus of the axis of vibration expressed in analytical form, which represents the geometrical locus of the eigenvector. A numerical example is used to clearly illustrate the vibration phenomena of an optical pick-up used in an information storage device.

Vibration analysis of single rigid-body systems having planes of symmetry

2002 ◽  
Vol 216 (6) ◽  
pp. 629-641 ◽  
Author(s):  
B J Dan ◽  
Y J Choi
Keyword(s):  
Rigid Body ◽  
Frequency Response ◽  
Vibration Analysis ◽  
Design Method ◽  
Dimensional Space ◽  
Information Storage ◽  
Applied Force ◽  
Mode Shapes ◽  
Storage Device ◽  
Geometrical Approach

By taking a geometrical approach to vibration analysis, the vibration mode shapes of a single rigid body may be better understood. From the geometrical point of view, the eigenvectors represent repetitive twisting motions on the axes of vibrations in a three-dimensional space. The frequency response can be expressed by a scalar multiple of the axis of vibration in Plücker's axis coordinates, which is the reciprocal product of the axis of vibration and applied force. The geometrical interpretation of the frequency response provides the design methodology to eliminate the undesired peaks. The methodology involves making the mutual moment of the axis of vibration and applied force become zero or adjusting the location of the axes of vibrations to the sensing point. In order to implement this geometrical design method, an optical pick-up used in the information storage device has been utilized. Some numerical results show design improvement concerning the frequency response of the pick-up device and thereby the validity of the design technique.

An Investigation of the Structural Dynamics of a Racing Yacht

1999 ◽  
Author(s):  
Frederic Louarn ◽  
Pandeli Temarel
Keyword(s):  
Dynamic Properties ◽  
Three Dimensional ◽  
Element Model ◽  
Point Of View ◽  
Mode Shapes ◽  
Function Technique ◽  
Operational Conditions ◽  
Principal Coordinates ◽  
Modal Summation ◽  
Structural Responses

The dynamic behaviour of a WOR 60 is investigated using three dimensional hydroelasticity theory. Global structural responses (e.g. stresses) in waves are obtained corresponding to the upright as well as to the more realistic heeled sailing configurations, revealing the connection between the ballast keel and the hull as being a critical area of the structure. For the "dry hull" analysis, a global finite element model has been developed, incorporating the hull and deck shell, the internal structure, the ballast keel and the rig together with rigging loads. The modular nature of the model has been used to assess the relative influence of each of the aforementioned components upon the required characteristic dynamic properties (e.g. natural frequencies and principal mode shapes). Regarding the "wet hull" analysis, a three dimensional Green's function technique, using pulsating sources distributed over the wetted surface, provides a numerical solution to the case of the yacht sailing in regular waves at arbitrary heading. Principal coordinates for the rigid body motions and flexible distortions of interest are evaluated and the latter are used to obtain the dynamic stresses in waves using modal summation. This paper will describe the modelling techniques used and discuss the applicability / limitations of hydroelasticity theory regarding this type of structures in the light of the results obtained for the upright and heeled operational conditions, as well as from the point of view of design aspects such as "L" and "T" keel configurations. The ABS design criteria will provide a practical reference for comparing the results from the dynamic analysis.

scholarly journals Platonic solids generate their four-dimensional analogues

2013 ◽  
Vol 69 (6) ◽  
pp. 592-602 ◽  
Author(s):  
Pierre-Philippe Dechant
Keyword(s):  
Coxeter Groups ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Root Systems ◽  
Point Of View ◽  
Platonic Solids ◽  
Four Dimensions ◽  
Potential Applications ◽  
Regular Convex ◽  
Quasi Crystals

This paper shows how regular convex 4-polytopes – the analogues of the Platonic solids in four dimensions – can be constructed from three-dimensional considerations concerning the Platonic solids alone.Viathe Cartan–Dieudonné theorem, the reflective symmetries of the Platonic solids generate rotations. In a Clifford algebra framework, the space of spinors generating such three-dimensional rotations has a natural four-dimensional Euclidean structure. The spinors arising from the Platonic solids can thus in turn be interpreted as vertices in four-dimensional space, giving a simple construction of the four-dimensional polytopes 16-cell, 24-cell, theF4root system and the 600-cell. In particular, these polytopes have `mysterious' symmetries, that are almost trivial when seen from the three-dimensional spinorial point of view. In fact, all these induced polytopes are also known to be root systems and thus generate rank-4 Coxeter groups, which can be shown to be a general property of the spinor construction. These considerations thus also apply to other root systems such as A_{1}\oplus I_{2}(n) which induces I_{2}(n)\oplus I_{2}(n), explaining the existence of the grand antiprism and the snub 24-cell, as well as their symmetries. These results are discussed in the wider mathematical context of Arnold's trinities and the McKay correspondence. These results are thus a novel link between the geometries of three and four dimensions, with interesting potential applications on both sides of the correspondence, to real three-dimensional systems with polyhedral symmetries such as (quasi)crystals and viruses, as well as four-dimensional geometries arising for instance in Grand Unified Theories and string and M-theory.

scholarly journals Three Dimensional Vibration Analysis of a Class of Traction-Free Solid Elastic Bodies with an Eccentric Cavity

2012 ◽  
Vol 19 (6) ◽  
pp. 1341-1357 ◽  
Author(s):  
Seyyed M. Hasheminejad ◽  
Yaser Mirzaei
Keyword(s):  
Vibration Analysis ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Numerical Results ◽  
Mode Shapes ◽  
Computational Techniques ◽  
Published Data ◽  
Elastic Structures ◽  
Wide Range ◽  
Vibrational Characteristics

A three-dimensional elasticity-based continuum model is developed for describing the free vibrational characteristics of an important class of isotropic, homogeneous, and completely free structural bodies (i.e., finite cylinders, solid spheres, and rectangular parallelepipeds) containing an arbitrarily located simple inhomogeneity in form of a spherical or cylindrical defect. The solution method uses Ritz minimization procedure with triplicate series of orthogonal Chebyshev polynomials as the trial functions to approximate the displacement components in the associated elastic domains, and eventually arrive at the governing eigenvalue equations. An extensive review of the literature spanning over the past three decades is also given herein regarding the free vibration analysis of elastic structures using Ritz approach. Accuracy of the implemented approach is established through proper convergence studies, while the validity of results is demonstrated with the aid of a commercial FEM software, and whenever possible, by comparison with other published data. Numerical results are provided and discussed for the first few clusters of eigen-frequencies corresponding to various mode categories in a wide range of cavity eccentricities. Also, the corresponding 3D mode shapes are graphically illustrated for selected eccentricities. The numerical results disclose the vital influence of inner cavity eccentricity on the vibrational characteristics of the voided elastic structures. In particular, the activation of degenerate frequency splitting and incidence of internal/external mode crossings are confirmed and discussed. Most of the results reported herein are believed to be new to the existing literature and may serve as benchmark data for future developments in computational techniques.

scholarly journals GEOMETRICAL APPROACH TO LIGHT IN INHOMOGENEOUS MEDIA

2002 ◽  
Vol 17 (11) ◽  
pp. 1543-1558 ◽  
Author(s):  
P. PIWNICKI
Keyword(s):  
Electromagnetic Fields ◽  
Light Propagation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Flat Space ◽  
Dielectric Medium ◽  
Index Of Refraction ◽  
Geometrical Approach ◽  
Complex Formulation ◽  
Relativistic Approach

Electromagnetism in an inhomogeneous dielectric medium at rest is described using the methods of differential geometry. In contrast to a general relativistic approach the electromagnetic fields are discussed in three-dimensional space only. The introduction of an appropriately chosen three-dimensional metric leads to a significant simplification of the description of light propagation in an inhomogeneous medium: light rays become geodesics of the metric and the field vectors are parallel transported along the rays. The new metric is connected to the usual flat space metric diag[1,1,1] via a conformal transformation leading to new, effective values of the medium parameters [Formula: see text] and [Formula: see text] with [Formula: see text]. The corresponding index of refraction is thus constant and so is the effective velocity of light. Space becomes effectively empty but curved. All deviations from straight-line propagation are now due to curvature. The approach is finally used for a discussion of the Riemann–Silberstein vector, an alternative, complex formulation of the electromagnetic fields.

scholarly journals Analytical model and energy harvesting analysis of a vibrating slender rod with added tip mass in three-dimensional space

2021 ◽  
Author(s):  
Marek Borowiec ◽  
Marcin Bochenski ◽  
Grzegorz Litak ◽  
Andrzej Teter
Keyword(s):  
Energy Harvesting ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Electrical Circuit ◽  
Mode Shapes ◽  
3D Space ◽  
Harvesting Systems ◽  
Load Resistor ◽  
Tip Mass

AbstractIn the paper, a new 3D energy harvesting system is provided. This work discussed the Lagrange approach to derive the differential equations of motion in the case of energy harvesting systems. An electromechanical system consists of a mechanical resonator, a piezoelectric transducer and electrical circuit with the load resistor. A flexible slender rod clamped at the bottom and loaded by the tip mass is proposed as the resonator. Moving in the 3D space, it enables the system to avoid the gravitational potential barrier of the straight vertical shape in case of buckling. This paper investigates the response of the rod deflection and the root mean square power output of selected vibration mode shapes with an attached tip mass.

scholarly journals Ideals of Herzog–Northcott type

2011 ◽  
Vol 54 (1) ◽  
pp. 161-186 ◽  
Author(s):  
Liam O'Carroll ◽  
Francesc Planas-Vilanova
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Regular Sequence ◽  
Point Of View ◽  
Prime Ideals ◽  
Monomial Curves ◽  
Definition Of ◽  
Minimal Prime ◽  
Special Case ◽  
Three Dimensional Space

AbstractThis paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

Experimental research on cage dynamic characteristics of angular contact ball bearing

2019 ◽  
Vol 20 (2) ◽  
pp. 204
Author(s):  
Qingqing Li ◽  
Xiaoyang Chen ◽  
Tao Zhang ◽  
Shijin Chen ◽  
Jiaming Gu
Keyword(s):  
Dynamic Characteristics ◽  
Performance Test ◽  
Rotation Speed ◽  
Dimensional Space ◽  
Dynamic Performance ◽  
Three Dimensional ◽  
Axial Displacement ◽  
Mass Center ◽  
Angular Contact Ball Bearing ◽  
Controllable Motion

The dynamic performance and life of the precise bearing, even abnormal operation and early failure are affected directly by the complex and unstable motion of the cage. Based on the cage dynamic performance test device with controllable motion of inner and outer rings, respectively, the dynamic characteristics of the cage were studied under different rotating speeds and loads, while inner ring rotated with outer ring fixed and inner–outer rings rotated reversely. Then the trajectory of the cage mass center was drawn through experimental study. The three-dimensional space motions of cage reveal that, when only inner ring rotates, the trajectories of cage mass center are approximately circular under axial load, and the amplitude of the axial displacement raises with the increase of the rotation speeds. With the increase of radial loads, the cage mass center trajectories are shaking from a circle to a small area on the side of the bearing center. When the inner–outer rings rotate in the opposite direction, the rotation speed of the cage is greatly reduced, and the mass center trajectories of the cage oscillate irregularly on side of the bearing center. As the relative rotation speed of rings increases, the axial displacement fluctuation enlarges. With the increase of the radial loads, the axial fluctuation decreases.

scholarly journals Perceptually Valid Facial Expressions for Character-Based Applications

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
Ali Arya ◽  
Steve DiPaola ◽  
Avi Parush
Keyword(s):  
Facial Expression ◽  
Language Learning ◽  
Facial Expressions ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Point Of View ◽  
Mixed Emotions ◽  
Dynamic Facial Expressions ◽  
Emotion Model ◽  
Three Dimensional Space

This paper addresses the problem of creating facial expression of mixed emotions in a perceptually valid way. The research has been done in the context of a “game-like” health and education applications aimed at studying social competency and facial expression awareness in autistic children as well as native language learning, but the results can be applied to many other applications such as games with need for dynamic facial expressions or tools for automating the creation of facial animations. Most existing methods for creating facial expressions of mixed emotions use operations like averaging to create the combined effect of two universal emotions. Such methods may be mathematically justifiable but are not necessarily valid from a perceptual point of view. The research reported here starts by user experiments aiming at understanding how people combine facial actions to express mixed emotions, and how the viewers perceive a set of facial actions in terms of underlying emotions. Using the results of these experiments and a three-dimensional emotion model, we associate facial actions to dimensions and regions in the emotion space, and create a facial expression based on the location of the mixed emotion in the three-dimensional space. We call these regionalized facial actions “facial expression units.”

scholarly journals THE SANSEVERO CHAPEL, A THREE-DIMENSIONAL POINT OF VIEW

2019 ◽  
Vol XLII-2/W15 ◽  
pp. 299-304
Author(s):  
V. Cera ◽  
D. Marcos González ◽  
L. A. Garcia
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Point Of View ◽  
Two Dimensional ◽  
Dimensional Representation ◽  
Survey Techniques ◽  
Representation Systems ◽  
Representation Of Space ◽  
Special Relevance ◽  
Three Dimensional Space

<p><strong>Abstract.</strong> In this article, the importance of the three-dimensional survey in architectural spaces will be studied, taking special relevance in the study of the perception of perspective, since three-dimensional space would not be understood from a two-dimensional representation of space. The project aims to develop a comparison between the representation systems based on the automatic acquisition of various data by different 3D survey techniques. In particular, the document reports the results of an analysis based on the Sansevero Chapel in Naples.</p>

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