Path curvatures in three-dimensional constrained motion of a rigid body

1989 ◽  
Vol 24 (2) ◽  
pp. 73-81 ◽  
Author(s):  
J Hirschhorn
Keyword(s):  
Rigid Body ◽  
Three Dimensional ◽  
Constrained Motion

Biomechanical characteristics of C1–2 cable fixations

1996 ◽  
Vol 85 (2) ◽  
pp. 316-322 ◽  
Author(s):  
Curtis A. Dickman ◽  
Neil R. Crawford ◽  
Christopher G. Paramore
Keyword(s):  
Cyclic Loading ◽  
Translational Motion ◽  
Biomechanical Testing ◽  
Three Dimensional ◽  
Constrained Motion ◽  
Content Type ◽  
Physiological Range ◽  
Fixation Techniques ◽  
Cable Fixation ◽  
Fine Print

✓ The biomechanical characteristics of four different methods of C1–2 cable fixation were studied to assess the effectiveness of each technique in restoring atlantoaxial stability. Biomechanical testing was performed on the upper cervical spines of four human cadaveric specimens. Physiological range loading was applied to the atlantoaxial specimens and three-dimensional motion was analyzed with stereophotogrammetry. The load–deformation relationships and kinematics were measured, including the stiffness, the angular ranges of motion, the linear ranges of motion, and the axes of rotation. Specimens were nondestructively tested in the intact state, after surgical destabilization, and after each of four different methods of cable fixation. Cable fixation techniques included the interspinous technique, the Brooks technique, and two variants of the Gallie technique. All specimens were tested immediately after fixation and again after the specimen was fatigued with 6000 cycles of physiological range torsional loading. All four cable fixation methods were moderately flexible immediately; the different cable fixations allowed between 5° and 40° of rotational motion and between 0.6 and 7 mm of translational motion to occur at C1–2. The Brooks and interspinous methods controlled C1–2 motion significantly better than both of the Gallie techniques. The motion allowed by one of the Gallie techniques did not differ significantly from the motion of the unfixed destabilized specimens. All cable fixation techniques loosened after cyclic loading and demonstrated significant increases in C1–2 rotational and translational motions. The bone grafts shifted during cyclic loading, which reduced the effectiveness of the fixation. The locations of the axes of rotation, which were unconstrained and mobile in the destabilized specimens, became altered with cable fixation. The C1–2 cables constrained motion by shifting the axes of rotation so that C-1 rotated around the fixed cable and graft site. After the specimen was fatigued, the axes of rotation became more widely dispersed but were usually still localized near the cable and graft site. Adequate healing requires satisfactory control of C1–2 motion. Therefore, some adjunctive fixation is advocated to supplement the control of motion after C1–2 cable fixation (that is, a cervical collar, a halo brace, or rigid internal fixation with transarticular screws).

Measurement of Angular Acceleration of a Rigid Body Using Linear Accelerometers

1975 ◽  
Vol 42 (3) ◽  
pp. 552-556 ◽  
Author(s):  
A. J. Padgaonkar ◽  
K. W. Krieger ◽  
A. I. King
Keyword(s):  
Experimental Data ◽  
Rigid Body ◽  
Simple Procedure ◽  
Three Dimensional ◽  
Angular Acceleration ◽  
Theory And Practice ◽  
Body Segments ◽  
Impact Biomechanics ◽  
The Stability ◽  
Definition Of

The computation of angular acceleration of a rigid body from measured linear accelerations is a simple procedure, based on well-known kinematic principles. It can be shown that, in theory, a minimum of six linear accelerometers are required for a complete definition of the kinematics of a rigid body. However, recent attempts in impact biomechanics to determine general three-dimensional motion of body segments were unsuccessful when only six accelerometers were used. This paper demonstrates the cause for this inconsistency between theory and practice and specifies the conditions under which the method fails. In addition, an alternate method based on a special nine-accelerometer configuration is proposed. The stability and superiority of this approach are shown by the use of hypothetical as well as experimental data.

Mathematical aspects of molecular replacement. IV. Measure-theoretic decompositions of motion spaces

2017 ◽  
Vol 73 (5) ◽  
pp. 387-402 ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Sajdeh Sajjadi ◽  
Bernard Shiffman ◽  
Steven M. Zucker
Keyword(s):  
General Theory ◽  
Rigid Body ◽  
Three Dimensional ◽  
Molecular Replacement ◽  
Translation Group ◽  
Common Occurrence ◽  
Space Group ◽  
Space Groups ◽  
Relevant Case ◽  
The Subject

In molecular-replacement (MR) searches, spaces of motions are explored for determining the appropriate placement of rigid-body models of macromolecules in crystallographic asymmetric units. The properties of the space of non-redundant motions in an MR search, called a `motion space', are the subject of this series of papers. This paper, the fourth in the series, builds on the others by showing that when the space group of a macromolecular crystal can be decomposed into a product of two space subgroups that share only the lattice translation group, the decomposition of the group provides different decompositions of the corresponding motion spaces. Then an MR search can be implemented by trading off between regions of the translation and rotation subspaces. The results of this paper constrain the allowable shapes and sizes of these subspaces. Special choices result when the space group is decomposed into a product of a normal Bieberbach subgroup and a symmorphic subgroup (which is a common occurrence in the space groups encountered in protein crystallography). Examples of Sohncke space groups are used to illustrate the general theory in the three-dimensional case (which is the relevant case for MR), but the general theory in this paper applies to any dimension.

scholarly journals GEOMETRIC MOMENTUM IN THE MONGE PARAMETRIZATION OF TWO-DIMENSIONAL SPHERE

2013 ◽  
Vol 10 (03) ◽  
pp. 1220031 ◽  
Author(s):  
D. M. XUN ◽  
Q. H. LIU
Keyword(s):  
Quantum Mechanics ◽  
Laplace Operator ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Constrained Motion ◽  
Geometric Invariant ◽  
Gradient Operator ◽  
The Laplace Operator

A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = iℏ(δij - xixj/r2) rather than [xi, pj] = iℏδij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

The Volume of Motion: Introduction, Derivation, and an Application Comparing Various Spinal Fixation Devices

1993 ◽  
Vol 115 (1) ◽  
pp. 43-46 ◽  
Author(s):  
J. J. Crisco
Keyword(s):  
Rigid Body ◽  
Three Dimensional ◽  
Spinal Fixation ◽  
Dimensional Parameter ◽  
Directional Information ◽  
One Dimensional ◽  
Fixation Devices ◽  
Spinal Fixation Devices ◽  
Mathematical Derivation

Range of motion (ROM), the displacement between two limits, is one of the most common parameters used to describe joint kinematics. The ROM is a one-dimensional parameter, although the motion at many normal and pathological joints is three-dimensional. Certainly, the ROM yields vital information, but an overall measure of the three-dimensional mobility at a joint may also be useful. The volume of motion (VOM) is such a measure. The translational VOM is the volume defined by all possible ROMs of a point on a rigid body. The rotational VOM, although its interpretation is not as tangible as the translational VOM, is a measure of the three-dimensional rotational mobility of a rigid body. The magnitude of the VOM is proportional to mobility; the VOM is a scaler, which does not contain any directional information. Experimental determination of the VOM is not practical since it would require applying loads in an infinite number of directions. The mathematical derivation given here allows the VOM to be calculated, with the assumption of conservative elasticity, from the resultant displacements of three distinct load vectors of equal magnitude. An example of the VOM is presented in the comparison of the biomechanical stabilizing potential of various spinal fixation devices.

On Saint-Venant's Problem for Helicoidal Beams

2015 ◽  
Vol 83 (2) ◽  
Author(s):  
Shilei Han ◽  
Olivier A. Bauchau
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Fem Analysis ◽  
Original System ◽  
Form Solution ◽  
Hamiltonian Matrix ◽  
Jordan Structure ◽  
Rigid Body Motions

This paper proposes a novel solution strategy for Saint-Venant's problem based on Hamilton's formalism. Saint-Venant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant's solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed-form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed-form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.

scholarly journals Hamilton’s Equations With Euler Parameters for Rigid Body Dynamics Modeling

2004 ◽  
Vol 126 (1) ◽  
pp. 124-130 ◽  
Author(s):  
Ravishankar Shivarama ◽  
Eric P. Fahrenthold
Keyword(s):  
Rigid Body ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Potential Energy Function ◽  
Rigid Body Dynamics ◽  
Dynamics Modeling ◽  
Euler Parameters ◽  
Strongly Nonlinear ◽  
Body Dynamics ◽  
General Potential

A combination of Euler parameter kinematics and Hamiltonian mechanics provides a rigid body dynamics model well suited for use in strongly nonlinear problems involving arbitrarily large rotations. The model is unconstrained, free of singularities, includes a general potential energy function and a minimum set of momentum variables, and takes an explicit state space form convenient for numerical implementation. The general formulation may be specialized to address particular applications, as illustrated in several three dimensional example problems.

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