EMch 112H
Kinematic Analysis of an Airfoil Cutting Mechanism

Introduction

In this project, we will use what we have learned in studying rigid body kinematics to do a complete kinematic analysis of a mechanism designed to cut out wing or airfoil profiles. Mechanisms like the one we will analyze are called four bar linkages (the ground is considered to be the fourth bar) and have been used since the beginning of the industrial revolution in machinery of all types. You will see during this project how complex even simple looking mechanisms can be to analyze. In addition, you will see how complex we can make the path traced out by different parts of the mechanism.

The Mechanism

Figure 1 shows a diagram (drawn approximately to scale) of the mechanism we will analyze in this project.

Figure 1. The mechanism we will analyze in this project.

The red circle at point E in the mechanism shown above is the laser cutting device. The center of the device is the point at which the cutting occurs. The motor at point D drives the link CD at a constant angular velocity w₀ (omega sub zero). The angles shown define the position of the mechanism at any given time. Figure 2 shows a QuickTime movie of the motion.

Figure 2. The motion of the 4-bar linkage.

The dimensions of the links in the above mechanism are: AB = 1.346 m, BC = 0.9925 m, CD = 0.5025 m, BE = 2.479, AD = 1.200 m.

The Analysis

Your task is to use the rigid body kinematics ideas you have learned so far this semester to:

Find and plot phi and gamma as a function of theta for theta between 0 and 2 pi.

Find and plot the trajectory of the laser cutter (point E) as a function of the input angle theta (you should get the trajectory seen in the QuickTime movie).

Find and plot the trajectory of the center of mass of bar CE as a function of the input angle theta (you should get the trajectory seen in the QuickTime movie).

Find and plot the x-component of the velocity, y-component of the velocity, and the total speed of point E as a function of the input angle. Find the maximum value of the input angular velocity so that the speed of the cutting tool never exceeds 0.5 m/s. Use this maximum value of the input angular velocity for the rest of the project.

Find and plot the angular acceleration of bars AB and CE as a function of the input angle theta.

Find and plot the x-component of the acceleration, y-component of the acceleration, and the total acceleration (magnitude) of point E as a function of the input angle.

Find and plot the x-component of the acceleration, y-component of the acceleration, and the total acceleration (magnitude) of the center of mass of link CE as a function of the input angle.

Many of these results will be used for your last project of the semester, which will be the kinetic analysis of this mechanism.

Project Report

The project report for this one is easy -- simply turn in the Mathematica notebook you use to do the kinematic analysis and do the plots. In addition, attach any handwritten work used to derive the appropriate equations. Be sure to put comments in your notebooks that clearly state exactly what you are doing each step of the way.

[ESM] [ESM Faculty] [PSU Engineering] [Penn State]

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Prepared by Gary L. Gray.
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