Experimental Analysis of Rigid Body Motion. A Vector Method to Determine Finite and Infinitesimal Displacements From Point Coordinates

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
Álvaro Page ◽  
Helios de Rosario ◽  
Vicente Mata ◽  
Carlos Atienza
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Angular Displacement ◽  
Linear Equations ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Simple Method ◽  
Vector Method ◽  
Displacement Analysis ◽  
Finite Displacements

This paper presents a vector method for measuring rigid body motion from marker coordinates, including both finite and infinitesimal displacement analyses. The common approach to solving the finite displacement problem involves the determination of a rotation matrix, which leads to a nonlinear problem. On the contrary, infinitesimal displacement analysis is a linear problem that can be easily solved. In this paper we take advantage of the linearity of infinitesimal displacement analysis to formulate the equations of finite displacements as a generalization of Rodrigues’ formula when more than three points are used. First, for solving the velocity problem, we propose a simple method based on a mechanical analogy that uses the equations that relate linear and angular momenta to linear and angular velocities, respectively. This approach leads to explicit linear expressions for infinitesimal displacement analysis. These linear equations can be generalized for the study of finite displacements by using an intermediate body whose points are the midpoint of each pair of homologous points at the initial and final positions. This kind of transformation turns the field of finite displacements into a skew-symmetric field that satisfies the same equations obtained for the velocity analysis. Then, simple closed-form expressions for the angular displacement, translation, and position of finite screw axis are presented. Finally, we analyze the relationship between finite and infinitesimal displacements, and propose vector closed-form expressions based on derivatives or integrals, respectively. These equations allow us to make one of both analyses and to obtain the other by means of integration or differentiation. An experiment is presented in order to demonstrate the usefulness of this method. The results show that the use of a set of markers with redundant information (n>3) allows a good accuracy of measurement of kinematic variables.

A Dual-Vector Method to Derive the Equivalent Screw of Two Successive Finite Screw Displacements for Line Segments

2020 ◽  
Vol 12 (4) ◽  
Author(s):  
Zetao Yu ◽  
Kwun-Lon Ting
Keyword(s):  
Rigid Body ◽  
Line Segment ◽  
Screw Theory ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Vector Method ◽  
Dual Vector ◽  
Similar Relation ◽  
Vector Algebra ◽  
Illustrative Purpose

Abstract For finite rigid body motion, every two successive screw displacements can be represented by one equivalent screw displacement. However, such phenomenon should not be considered naturally to be valid for incompletely specified displacements (ISDs). There is neither a precise statement for such phenomenon nor an understanding of its range of validity within ISD, such as line segment displacements. As one of the main contributions in this paper, based on dual vector algebra and screw theory, an algorithm is provided to prove the existence of the subset within the scope of the line segment motion, which expresses the similar relation as shown in finite rigid body motion. A numerical example is presented for illustrative purpose.

Estimation of Size and Depth of Debonds in Laminates From Holographic Interferometry

1992 ◽  
Vol 114 (2) ◽  
pp. 127-131 ◽  
Author(s):  
H. M. Shang ◽  
E. M. Lim ◽  
K. B. Lim
Keyword(s):  
Rigid Body ◽  
Holographic Interferometry ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Accurate Assessment ◽  
Plate Model ◽  
Double Exposure ◽  
Testing Technique ◽  
Simple Method ◽  
Exposure Holography

Double-exposure holography has been shown to be a useful nondestructive testing technique for the detection of debonds in laminates when subjected to vacuum stressing. The size of a debond is estimated from the boundary containing anomalous fringes and subsequently, its depth is estimated from these fringes using a thin-plate model. Rigid-body motion is frequently present during testing, causing ambiguity in the identification of the shape of the debond and hence difficulty in the estimation of its size and depth. In this paper, a simple method is presented which isolates the effects of rigid-body motion so that accurate assessment of the detected debonds can be achieved. Also, the use of a circle to outline the ambiguous boundary is found suitable for square debonds.

A planar reconfigurable linear rigid-body motion linkage with two operation modes

2014 ◽  
Vol 228 (16) ◽  
pp. 2985-2991 ◽  
Author(s):  
Guangbo Hao ◽  
Xianwen Kong ◽  
Xiuyun He
Keyword(s):  
Rigid Body ◽  
Single Mode ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Reference Guide ◽  
Revolute Joints ◽  
Operation Modes ◽  
Straight Line ◽  
Motion Stage ◽  
Motion Mode

A planar reconfigurable linear (also rectilinear) rigid-body motion linkage (RLRBML) with two operation modes, that is, linear rigid-body motion mode and lockup mode, is presented using only R (revolute) joints. The RLRBML does not require disassembly and external intervention to implement multi-task requirements. It is created via combining a Robert’s linkage and a double parallelogram linkage (with equal lengths of rocker links) arranged in parallel, which can convert a limited circular motion to a linear rigid-body motion without any reference guide way. This linear rigid-body motion is achieved since the double parallelogram linkage can guarantee the translation of the motion stage, and Robert’s linkage ensures the approximate straight line motion of its pivot joint connecting to the double parallelogram linkage. This novel RLRBML is under the linear rigid-body motion mode if the four rocker links in the double parallelogram linkage are not parallel. The motion stage is in the lockup mode if all of the four rocker links in the double parallelogram linkage are kept parallel in a tilted position (but the inner/outer two rocker links are still parallel). In the lockup mode, the motion stage of the RLRBML is prohibited from moving even under power off, but the double parallelogram linkage is still moveable for its own rotation application. It is noted that further RLRBMLs can be obtained from the above RLRBML by replacing Robert’s linkage with any other straight line motion linkage (such as Watt’s linkage). Additionally, a compact RLRBML and two single-mode linear rigid-body motion linkages are presented.

Computation of Nonlinear Response of Hydraulically Actuated Structures

1997 ◽  
Author(s):  
T. D. Burton ◽  
C. P. Baker ◽  
J. Y. Lew
Keyword(s):  
Rigid Body ◽  
Flexible Structures ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Flexible Beam ◽  
Test Bed ◽  
Ode Models ◽  
Hydraulically Actuated ◽  
Pressure Dynamics ◽  
Low Order

Abstract The maneuvering and motion control of large flexible structures are often performed hydraulically. The pressure dynamics of the hydraulic subsystem and the rigid body and vibrational dynamics of the structure are fully coupled. The hydraulic subsystem pressure dynamics are strongly nonlinear, with the servovalve opening x(t) providing a parametric excitation. The rigid body and/or flexible body motions may be nonlinear as well. In order to obtain accurate ODE models of the pressure dynamics, hydraulic fluid compressibility must generally be taken into account, and this results in system ODE models which can be very stiff (even if a low order Galerkin-vibration model is used). In addition, the dependence of the pressure derivatives on the square root of pressure results in a “faster than exponential” behavior as certain limiting pressure values are approached, and this may cause further problems in the numerics, including instability. The purpose of this paper is to present an efficient strategy for numerical simulation of the response of this type of system. The main results are the following: 1) If the system has no rigid body modes and is thus “self-centered,” that is, there exists an inherent stiffening effect which tends to push the motion to a stable static equilibrium, then linearized models of the pressure dynamics work well, even for relatively large pressure excursions. This result, enabling linear system theory to be used, appears of value for design and optimization work; 2) If the system possesses a rigid body mode and is thus “non-centered,” i.e., there is no stiffness element restraining rigid body motion, then typically linearization does not work. We have, however discovered an artifice which can be introduced into the ODE model to alleviate the stiffness/instability problems; 3) in some situations an incompressible model can be used effectively to simulate quasi-steady pressure fluctuations (with care!). In addition to the aforementioned simulation aspects, we will present comparisons of the theoretical behavior with experimental histories of pressures, rigid body motion, and vibrational motion measured for the Battelle dynamics/controls test bed system: a hydraulically actuated system consisting of a long flexible beam with end mass, mounted on a hub which is rotated hydraulically. The low order ODE models predict most aspects of behavior accurately.

Structures of the Phosphazenes [ClC(NPCl3)2]+PCl6 − and [CH3C(NPCl3)2]+SbCl6 − at 90 K

1997 ◽  
Vol 53 (6) ◽  
pp. 953-960 ◽  
Author(s):  
F. Belaj
Keyword(s):  
Rigid Body ◽  
Motion Analysis ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Torsion Angles ◽  
Ionic Compounds ◽  
Intramolecular Motions

The asymmetric units of both ionic compounds [N-(chloroformimidoyl)phosphorimidic trichloridato]trichlorophosphorus hexachlorophosphate, [ClC(NPCl3)2]+PCl^{-}_{6} (1), and [N-(acetimidoyl)phosphorimidic trichloridato]trichlorophosphorus hexachloroantimonate, [CH3C(NPCl3)2]+SbCl^{-}_{6} (2), contain two formula units with the atoms located on general positions. All the cations show cis–trans conformations with respect to their X—C—N—P torsion angles [X = Cl for (1), C for (2)], but quite different conformations with respect to their C—N—P—Cl torsion angles. Therefore, the two NPCl3 groups of a cation are inequivalent, even though they are equivalent in solution. The very flexible C—N—P angles ranging from 120.6 (3) to 140.9 (3)° can be attributed to the intramolecular Cl...Cl and Cl...N contacts. A widening of the C—N—P angles correlates with a shortening of the P—N distances. The rigid-body motion analysis shows that the non-rigid intramolecular motions in the cations cannot be explained by allowance for intramolecular torsion of the three rigid subunits about specific bonds.

Forward Displacement Analysis of Nine-Link Barranov Truss Based on Anti-Control of Chaos Newton Downhill Method

2011 ◽  
Vol 230-232 ◽  
pp. 749-753
Author(s):  
You Xin Luo ◽  
Ying Yang
Keyword(s):  
Body Motion ◽  
Control Of Chaos ◽  
Vector Method ◽  
Forward Displacement ◽  
Displacement Analysis ◽  
Real Solutions ◽  
Chaotic Sequences ◽  
Motion System ◽  
Constrained Equations ◽  
Forward Displacement Analysis

The anti-control of chaos Newton downhill method finding all real solutions of nonlinear equations was proposed and the forward displacement analysis on the 25th nine-link Barranov truss was completed. Four constrained equations were established by vector method with complex numbers according to four loops of the mechanism and four supplement equations were also established by increasing four variables and the relation of sine and cosine function. The established eight equations are that of forward displacement analysis of the mechanism. Combining Newton downhill method with chaotic sequences, anti-control of chaos Newton downhill method based on utilizing anti-control of chaos in body motion system to obtain locate initial points to find all real solutions of the nonlinear questions was proposed. The numerical example was given.The result shows that all real solutions have been quickly obtained, and it proves the correctness and validity of the proposed method.

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