MEGN 536 Computational Biomechanics Rotations for Rigid Body Kinematics Prof. Anthony Petrella

Knee (or any joint) Biomechanics  You are all MATLAB ninja or sous-ninja now  So, let’s start thinking about knee biomechanics…  Joint motion (kinematics) and forces (kinetics) are the foundation for defining design requirements  How do we express these quantities?  Let’s consider an example…

Introducing Abaqus Knee Simulator (AKS)  AKS has automation to help designer…  Test knee implant laxity  Test knee implant contact mechanics  Test knee implant wear performance  To run it…  Create a working folder in C:\scratch  Right click in folder and “open command prompt here”  At command prompt… >> abaqus kneeapp

Run Abaqus Knee Model  Download Abaqus knee model from project page, unzip (on hard drive!)  Follow PDF instructions to run solution  View an animation of a gait cycle… 1.Color Code  by Part instances

Run Abaqus Knee Model  Download Abaqus knee model from project page, unzip (on hard drive!)  Follow PDF instructions to run solution  View an animation of a gait cycle… 1.Color Code  by Part instances 2.Options  Common…  Feature edges

Run Abaqus Knee Model  Download Abaqus knee model from project page, unzip (on hard drive!)  Follow PDF instructions to run solution  View an animation of a gait cycle… 1.Color Code  by Part instances 2.Options  Common…  Feature edges 3.Result  Active Steps/Frames…  deactivate “compress”

Run Abaqus Knee Model  Download Abaqus knee model from project page, unzip (on hard drive!)  Follow PDF instructions to run solution  View an animation of a gait cycle… 1.Color Code  by Part instances 2.Options  Common…  Feature edges 3.Result  Active Steps/Frames…  deactivate “compress” 4.Options  Animation…  make Frame Rate about 75%

Run Abaqus Knee Model  Download Abaqus knee model from project page, unzip (on hard drive!)  Follow PDF instructions to run solution  View an animation of a gait cycle… 1.Color Code  by Part instances 2.Options  Common…  Feature edges 3.Result  Active Steps/Frames…  deactivate “compress” 4.Options  Animation…  make Frame Rate about 75% 5.Use Ctrl+Alt+Left mouse to rotate view 6.Animate  Time History  Question: What’s the best way to express 6 DOF kinematics and forces for this knee?

Rigid Body Motion  It is standard to describe rigid body motion by rotations and translations of a reference frame fixed in body – relative to a fixed global frame  For the knee, we will measure motion of the tibia relative to the femur  Femur will act like fixed global and tibia will be moving (Frey et al., 2006) Right Leg X Z Y z y x

Rigid Body Rotations  We will begin to consider rigid body motion with rotations only (translations are easy)  We’ll define the global fixed frame with capital letters and the moving local frame with lowercase letters

Rigid Body Motion

 Collecting the three dot products into a single expression gives…  Rotation matrix Rz is called an “orthonormal” rotation matrix because the rows are orthogonal unit vectors  Notice the rows of Rz are the unit vectors defining the local frame L  Rz also has the property…

Rigid Body Motion