3.2 - Project Prototype
We decided to implement a cam-follower mechanism for our music player, where 3 cams would be designed such that the varying radius of each cam would cause the mechanisms to drop a mallet on a key at precise times.
The overall goal of our prototype was to prove that with a cam, we can mimic a malloc striking a key on a xylophone.
Kinematic Analysis
We broke down our mechanism into two separate 4-bar mechanisms, pictured red and blue below.
We considered l2 as a link with variable length to represent the varying radius of the cam, from minimum radius to maximum, while theta_2 remained constant.
Position Analysis
Solving for first 4-bar (red):
Input: l1, l2_vector, l3, l4, theta_2
Output: theta3, theta4
We took a trigonomic approach towards solving the first four-bar. We considered l3 as a fixed ternary link with l2, as shown below, and solved for the law of cosines.
γ = arccos(a^2 + l1^2 - l2_vector^2 / 2*a*l1), where a was measured as 2.341 inches.
theta_4 = β + γ, where β was measured as 22 deg.
In our design with the cam and follower system, the cam will gradually increase in length from the pin to contact point of the follower until it hits our desired max value, and then have a drop in length, represented by l2. This in turn will cause a gradual increases of theta 4, and a sharp decrease in theta 4 to produce a striking motion in our mechanism. The graph below animates the relationship between l2 length changes (the cam to follower length changing) and theta 4 values. Step indices 0-100 relate to the increasing l2 values (from the minimum to maximum values as shown above), and the step values 100-110 relate to the decreasing l2 values (from the maximum to minimum values).
Solving for second 4-bar (blue):
alpha_global = 23degrees (angle between red to blue coordinate system)
theta_2 = theta_4 - alpha_global
Input: l1, l2, l3, l4, theta_2
Output: theta3, theta4
We solved the system as an open configuration 4-bar.
Plotting point P in the coordinate system of the second 4-bar
R_p = R_a + R_ap
R_a = l2*(cos(theta_2) + j*sin(theta_2))
R_a_real = l2*cos(theta_2)
R_a_imaginary = l2*sin(theta_2)
R_ap = l_ap*(cos(theta3) + j*sin(theta3))
R_ap_real = l_ap*cos(theta3)
R_ap_imaginary = l_ap*sin(theta3)
RP_real = R_a_real + R_ap_real
RP_imaginary = R_a_imaginary + R_ap_imaginary
Plotting point P in coordinate system of first 4-bar (global)
R_a_globalX = RP_real*cos(alpha_global) - RP_imaginary*sin(alpha_global)
R_a_globalY = RP_real*sin(alpha_global) + RP_imaginary*cos(alpha_global)
Overall, we verified that the trajectory of Point P will take on a curved shape with a quick decline, such that it is able to strike the xylophone key as expected. Our analysis shows that this is feasible.
Gruebler’s Equation
F = 3(n-1)-2L-H
F = number of degrees of freedom
n = total number of links in the mechanism
L = total number of lower pairs (1 DOF such as pins and sliding joints)
H = total number of higher pairs (2 DOF such as cam and gear joints)
For our mechanism:
F = 3(4-1)-2(3)-1 = 2 for the red 4-bar
F = 3(4-1)-2(4)-0 = 1 for the blue 4-bar
Grashof’s Law
For the red 4-bar:
L1 = 3in
L2 = changes from 0.4in → 1.75in
L3 = 1.75in
L4 = 1.75in
S + L ? P + Q
1.75 + 3 ? 1.75 + 1.75
4.75 > 3.5 ->Non-Grashof
For the blue 4-bar:
L1 = 4.8in
L2 = 3.25in
L3 = 3.098in (full is 7in)
L4 = 1.5in
S + L ? P + Q
1.5 + 4.8 ? 3.25 + 3.098
6.3 > 6.348 ->Non-Grashof
Iteration Documentation
Initial Iteration
After sufficient brainstorming on how to deliver a linkage system that displays a motion applicable to play a xylophone, we decided to implement four revolute joints in series attached to a driving higher pair joint: AKA a CAM system. Our initial iteration was designed relatively to scale to the MotionGen design in Figure 2. It is important to note that in MotionGen, there are no CAM components, so we used a piston to reflect the variable lengths that the system would experience with a CAM. To get a visual idea of our selected design, we used a cereal box to quickly mock up a prototype to prove the articulations of each joint. At this iteration, we were not concerned with the effects of different linkage lengths because the cereal box material was too flimsy to maneuver accurately. We designated our second prototype iteration to troubleshoot and test all of the variables of different lengths and cam positions.
Secondary Iteration
We decided to mock up our second iteration prototype using ¼” wood slabs due to the accessibility, cost, and durability of the material. We laser cut all of the lever arms and the baseboard out of the wood and 3d printed our cam and follower components. We made sure to print extra levers and to add extra joint holes on certain lever arms so that we could test different iterations within the same prototype. Video 2 is the depiction of our final iteration because it had the best performance of the motion we were looking for out of the four iterations of tests we conducted with the same prototype.
Videos 4-6 depict different iterations of our prototype and with careful inspection of the full range of motion of the rightmost link, it is clear to see that Iteration 2.1 was best suited for hitting a xylophone.
BOM