scholarly journals Appendix II: Orientation of a Body-fixed Coordinate System

2018 ◽  
pp. 683-685
Keyword(s):  
Coordinate System ◽  
Body Fixed Coordinate System

scholarly journals Zur exakten Behandlung von kollektiven Rotationen Teil A: Theorie / On the Exact Treatment of Collective Rotations Part A: Theory

1973 ◽  
Vol 28 (2) ◽  
pp. 206-215
Author(s):  
Hanns Ruder
Keyword(s):  
Coordinate System ◽  
Perturbation Expansion ◽  
Rotational Energy ◽  
The Body ◽  
System A ◽  
N Body Problem ◽  
Definition Of ◽  
Fixed System ◽  
Body Fixed Coordinate System ◽  
Groundstate Energy

Basic in the treatment of collective rotations is the definition of a body-fixed coordinate system. A kinematical method is derived to obtain the Hamiltonian of a n-body problem for a given definition of the body-fixed system. From this exact Hamiltonian, a consequent perturbation expansion in terms of the total angular momentum leads to two exact expressions: one for the collective rotational energy which has to be added to the groundstate energy in this order of perturbation and a second one for the effective inertia tensor in the groundstate. The discussion of these results leads to two criteria how to define the best body-fixed coordinate system, namely a differential equation and a variational principle. The equivalence of both is shown.

scholarly journals Numerical Analysis of Second-Order Mean Wave Forces by a Stabilized Higher Order Boundary Element Method

2018 ◽  
Author(s):  
Yan-Lin Shao
Keyword(s):  
Free Surface ◽  
Boundary Element Method ◽  
Coordinate System ◽  
Boundary Element ◽  
Matrix Equation ◽  
Wave Structure ◽  
Surface Conditions ◽  
Structure Interaction ◽  
Wave Structure Interaction ◽  
Body Fixed Coordinate System

A stabilized Higher-Order Boundary Element Method (HOBEM) based on cubic shape functions is presented to solve the linear wave-structure interaction with the presence of steady or slowly varying velocities. The m-terms which involves second derivatives of local steady flow are difficult to calculate accurately on structure surfaces with high curvatures. They are also not integrable at the sharp corners. A formulation of the Boundary Value Problem (BVP) in a body-fixed coordinate system is thus adopted, which avoids the calculation of the m-terms. The use of body-fixed coordinate system also avoid the inconsistency in the traditional perturbation method when 2nd order slowly-vary motions are larger than the linear motions. The stabilized numerical method presented in this paper is based on streamline integration and biased differencing scheme along the streamlines. The presence of convective terms in the kinematic and dynamic free surface conditions will lead to instable solution if the explicit method is used. Thus a fully implicit scheme is used in this paper for the time integration of kinematic and dynamic free surface conditions. In an implicit scheme, solution of an additional matrix equation is normally required due to the fact that the presence of convective terms are approximated using the variables at current time step rather than the previous time steps only. A method that avoids solving such matrix equation is presented in this paper, which will reduce the computational efforts in the implicit method. The methodology is applicable on unstructured meshes. It can also be used in general second order wave-structure interaction analysis with presence of steady or slowly-varying velocities.

2-D Inverse Dynamics Knee Model: Aligning Anatomical Knee Model With Knee Extension Kinematic Data Using Ligament Forces

2018 ◽  
Author(s):  
Jose Mario Salinas ◽  
Dumitru I. Caruntu
Keyword(s):  
Experimental Data ◽  
Coordinate System ◽  
Inverse Dynamics ◽  
Knee Extension ◽  
The Body ◽  
Body Segment ◽  
Center Of Rotation ◽  
Axis Of Rotation ◽  
Instantaneous Center ◽  
Body Fixed Coordinate System

This paper deals with aligning knee geometrical anatomical data with kinematic data from experimental work in order to develop a two-dimensional inverse dynamics anatomical model of human knee. Motion capture cameras were used to collect the experimental data for a knee extension exercise. Reflective markers were placed on the subjects’ skin during the experiment. In this model, joints such as hip, knee, and ankle are represented by axes of rotation. These axes are determined by calculating the relative instantaneous center of rotation of one body segment with respect to an adjacent body segment. Body-fixed coordinate systems were defined using three reflective markers attached to the subject. The origin of each body fixed-coordinate system was located between the three markers on that body segment, the body-fixed x-axis was pointing towards the marker on the lateral side of the body segment, and the body-fixed y-axis fell on the same plane as the three reflective markers on the body segment. Moreover, the axis of rotation that represents the hip was determined by calculating the instantaneous center of rotation of reflective markers located on the pelvis with respect to a body fixed coordinate system on the thigh. The axis of rotation on the knee was determined by calculating the instantaneous center of rotation of reflective markers on the shin (tibia) with respect to the body-fixed coordinate system on the thigh (femur). The axis of rotation on the ankle was determined by calculating the instantaneous center of rotation of reflective markers on the shin with respect to a body-fixed coordinate system on the foot. Bone anatomical geometries of femur and tibia were represented mathematically as polynomials and superimposed over the experimental data. This was done by matching the center of rotation from experimental data with the geometric center of the femoral condyle. This is necessary for estimating the insertions/origins of knee ligaments. These ligaments are modeled as nonlinear elastic springs. Furthermore, ligaments were divided into separate fiber bundles. Both the posterior and anterior cruciate ligaments were divided into an anterior and posterior fiber bundle. The cruciate ligament forces for both exercises are discussed in this paper.

scholarly journals Numerical Analysis of Second-Order Mean Wave Forces by a Stabilized Higher-Order Boundary Element Method

2019 ◽  
Vol 141 (5) ◽  
Author(s):  
Yan-Lin Shao
Keyword(s):  
Free Surface ◽  
Boundary Element Method ◽  
Coordinate System ◽  
Matrix Equation ◽  
Wave Structure ◽  
Second Order ◽  
Surface Conditions ◽  
Structure Interaction ◽  
Wave Structure Interaction ◽  
Body Fixed Coordinate System

A stabilized higher-order boundary element method (HOBEM) based on cubic shape functions is presented to solve the linear wave-structure interaction with the presence of steady or slowly varying velocities. The m-terms which involve second derivatives of local steady flow are difficult to calculate accurately on structure surfaces with large curvatures. They are also not integrable at the sharp corners. A formulation of the boundary value problem in a body-fixed coordinate system is thus adopted, which avoids the calculation of the m-terms. The use of body-fixed coordinate system also avoids the inconsistency in the traditional perturbation method when the second-order slowly varying motions are larger than the first-order motions. A stabilized numerical method based on streamline integration and biased differencing scheme along the streamlines will be presented. An implicit scheme is used for the convective terms in the free surface conditions for the time integration of the free surface conditions. In an implicit scheme, solution of an additional matrix equation is normally required because the convective terms are discretized by using the variables at current time-step rather than that from the previous time steps. A novel method that avoids solving such matrix equation is presented, which reduces the computational efforts significantly in the implicit method. The methodology is applicable on both structured and unstructured meshes. It can also be used in general second-order wave-structure interaction analysis with the presence of steady or slowly varying velocities.

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