Automatic Dry Eraser: Report
Problem Statement
Motivation
Our team wanted to build a mechanism that could erase a whiteboard on its own. Many automatic whiteboard erasers have already been built with a simple design of a large eraser that slides horizontally as shown below on the left. The eraser slides from right to left to erase the entire board. But what if one wishes to only erase sections of the board? If these sections are not arranged in a column, then such a simple design fails. Next, we consider a mechanism with two prismatic sliders to move an eraser in both the x and y directions (below, on the right). However, this design requires two actuators. We seek to design a mechanism that can move point by point with only one actuator. Such a mechanism could be advanced in the future to include an additional actuator to press the eraser against the board.
("Automatic Whiteboard Cleaner". 2018) (“iBoardbot. The internet controlled whiteboard robot,” 2018)
Initial Design
Brainstorming & Initial Solution
Our group's initial concept was to create a mechanism that moved up and down after each revolution, but translated slightly in the orthogonal direction after each revolution as well. An optimal trajectory path is shown in the figure below.
We first thought to accomplish this with a six-bar linkage, an eight-bar linkage, or a cam-follower coupled with a quick return mechanism. After greater consideration, we opted to couple a quick return mechanism with a rack and pinion. That would be the simplest way to ensure that each rotation has an identical profile, yet translated in the x-direction. Using the same actuator to drive the pinion and the first link accomplishes this task elegantly. A preliminary sketch is shown below.
It is important to note that although our desired pathway is only an x-y relation, it does contain time history; as more time passes, each point is moved farther to the left (assuming the mechanism starts at the right and moves to the left). Thus, we must design our mechanism to move up and down as quickly as possible. Otherwise, movements will be elongated and stretched out. Next, we consider a quick return mechanism for our design and kinematic analysis.
(“Whitworth Quick Return Mechanism - YouTube,” 2018).
CAD Drawings & Simulations
Kinematic & Motion Analysis
Approach
The key to a quick return is to have a small change in input angle translate into a large translation of a single point. Unlike the Whitworth quick return mechanism, we will attempt a four-bar quick return mechanism. We will attempt to produce large changes in x with a small change in ϴ2.
We begin by writing a vector loop equation to solve for x. Then, we differentiate with respect to time to find the time derivative of x:
Next, we consider two important case scenarios in the revolution of ϴ2:
Since the velocity of point X is 0 in any case such that ϴ2 = ϴ3, we consider the mechanism to move point X in the following sequence:
ϴ2 = ϴ3: Begin to extend point X
ϴ2 ⟂ ϴ3: Point X is halfway through its extension
ϴ2 = ϴ3: Point X is fully extended. Begin to retract point X
ϴ2 ⟂ ϴ3: Point X is halfway through its retraction
ϴ2 = ϴ3: Point X is fully retracted. Extend point X again.
Given this approach, it is natural that we would attempt to optimize this problem by maximizing the velocity of X when ϴ2 ⟂ ϴ3.
Kinematic Analysis
To maximize the velocity of X when ϴ2 ⟂ ϴ3, we choose not to increase L2 or ⍵2, given that these factors are not unique to the position of interest. Instead, we are interested in a value hereby denoted as the velocity multiplication factor, which is equal to (cos(ϴ2) *tan(ϴ4 - ϴ2) + sin(ϴ2)). First, we determine arbitrary values of L2, L3, and L1 based on convenient geometry for our application. We will use L2 = 4, L3 = 8, and L1 = 6.44 (L1 is defined from a trigonometrical relationship, which is why it is an uneven value). Next, We can solve for ϴ4 given ϴ2. We define ϴ2 as a vector spanning from 0° to 90°. Now, all of our variables are defined, and we can look at a plot of velocity multiplication factor vs. ϴ2 via the MATLAB script velProf.m
The plot shows a minimum at 72°. Thus, this is the angle that ϴ2 should be at when ϴ2 ⟂ ϴ3. From here, we are curious in the geometry of ϴ4 of and L4 at this specific position. The function geomSolver solves for these values for the corresponding values of L2, L3, L1, and ϴ2. The function geomSolver has an additional input, L3O, which is the overhang length from the L3 - L4 pin joint. We will start with L3O = 4. The input
geomSolver(4, 8, 6.44, 4, 72) gives a value of L4 = 6.27 inches. We will use 6.25 inches for convenience sake. The function geomSolver also calls a function named oneRev, which shows the plot of point X after one untitled revolution. The plot is as shown below.
Now, we must tilt the plot to give us our desired up and down motion. The general trend of the plot is travelling up 4 and to the right 12. Initially, a line of best fit was used, but had to be disregarded. A line of best fit for such a plot would be biased towards the periods of low velocity, which take up a comparatively less total time than the high velocity sections. The trend corresponds to an angle of 18°. To make the quick return go down, then up, we must reflect it about the x axis (now -18°) and orient it with respect to the y axis. The total tilt, including additional tilt added to overcome the leftwards translation, is -69°. We make this our ϴ1 value. Now, we are ready for a plot of the eraser position over time. The script eraserPos accomplishes this for us, with a variable for the pitch radius of the pinion. Other variables are included but do not affect the graphical plot of the linkage.
The plot is good and minimizes white space considering the width of the eraser. However, the plot begins with a small amount of backwards motion (shown at the top right of the image). We seek to eliminate this by setting the initial angle to a y-position maximizing value. We then set the initial angle to 7° and achieve the following plot.
All geometric definitions are made according to the following drawing.
Manufacturing and Assembly
Bill of Materials
Item | Dimensions/Specifications | Quantity |
|---|---|---|
Dry Eraseboard | 17" x 23" Whiteboard | 1 |
Tension Clips | 1/4" E clips 98408A120 McMaster | 10 |
8mmx16mmx5mm | 10 | |
25 mm diameter | 4 | |
1/4 inch x 3 inch x 2 ft | 1 |
Wood Pillars 3/4 inch x 3/4 inch x 96 inch2 - Qty 1
Tetrix DC Motor 12 Volt, 152 rpm, 320 oz-in torque Qty 1
Driven Voltage 4.8～35V; Logic supply current ≤36mA
3D Printing
Our initial plan for manufacturing our linkage system was to laser cut them out of an acrylic sheet. Our 3D printed links were to serve as a prototype to determine how closely our linkage system trajectory matched our theoretical trajectory.
However, time constraints involving laser-cut training, as well as the software it required prompted us to keep our 3D printed links. Also, the feedback from slider-crank project indicated several groups experienced manufacturing issues due the cut angle produced by the laser. Consequently, many groups experienced difficulty press-fitting bearings and were forced to extend time drilling.
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Manufacturing Progression
For our initial prototype, we 3D printed links with convenient (not fully optimized) lengths. Although the path of the links was not as desired, we were more interested in constructing a system of links that traveled through a full rotation. The carriage (structure that mounts motor to board and allows it to glide on casters) was built from metal, and the rack and pinion were as well 3D printed. All parts were mounted onto a non-magnetic whiteboard, which was challenging to establish pressure with. Below is an image of the initial carriage built.
In building our final prototype, the main changes we made were to 3D print the rack and pinion thicker for better meshing, as well as 3D printing the links at the correct lengths. Small changes were made to the carriage to adjust to new geometry. Lastly, everything was now mounted to a magnetic whiteboard. We were able to place the eraser in the correct location, and put magnets in the eraser to maintain pressure against the whiteboard. Some issues developed in the final design with the 3D printed parts. Since the initial prototype did not involve a true quick return, the links did not experience the acceleration and force that the final prototype did. These imposed a great deal of deflection on flexible 3D printed parts. Additionally, a key supporting part in the final design was cantilevered, and this caused a great deal of vibration and deflection. In the future, links should be made from a sturdier material like aluminum, and the cantilever should be eliminated.