Group 1 2012 Gate 4
In this gate, our group performed a complete reassembly of the piano. Using our deconstruction guide we created in gate 2, we were able to reverse the steps, working backwards until the piano was reassembled. In this section we also analyzed two of the mechanisms within the piano, including the physics and equations behind their purposes. Using the knowledge that we gathered from the project so far, we also introduced three revisions that could be made to the piano on a system level, that would solve some key issues, as well as provide a benefit to the consumer when considering the GSEE factors.
Cause for Corrective Action
The decision to decentralize the work involved in completing Gate 3 by assigning each group member a specific task and having no group meetings proved to be highly effective in minimizing the total amount of time spent on the project while still providing an acceptable degree of quality. Given the fractured and busied nature of everyone in the group\'s schedule during this project gate, we will continue to use this method to complete Gate 4.
As a result of our knowledge of its deconstruction, and the fact that the reassembly of the piano essentially proceeds in reverse order from its deconstruction, piano reassembly proceeded much more smoothly and efficiently than its deconstruction. Given the relative simplicity of each component and each component interaction, we have defined the complexity of each task as the approximate number of minutes required for its completion.
The majority of motion in a piano relies on interconnected levers to transmit energy or control to the strings. However, the initially simple motions of each individual part interact to create a very controllable set of movements necessary for piano playing. In the action and elsewhere, there are push rods and pull rods that effectively create our-bar mechanisms.
- Mechanism name: Modified four-bar linkage
- Purpose: The slow, short range-of-motion movement of keys translates to fast, wider-angle movement of the hammer. Then the linkage collapses and allows the hammer to retract.
- Implementation: The key is connected to a central lever in the action (r₄ in the linkage diagram). The lever has a push rod (r₃) that acts as a couple to the hammer base (r₂). The push bar falls out of place at the top of the key\'s motion and allows the hammer to return under gravity. The mounting hardware is r₁.
- Governing equations: 
Manipulating the equations above and substituting r₁=9cm, r₂=1.2cm, r₃=7cm, and r₄=3.6cm yields the following relationship:
The following graph shows the relationship where φ₄ is on the x-axis and φ₂ is on the y-axis. Angles are in radians. Since the linkage has a limited range of motion, the function gives a complex number outside of the range of motion.
Note how the slope is very steep from approximately 2 radians to 2.7 radians—due to the domain of arccos and the direction of the angles, the range of motion appeared in the fourth quadrant instead of the first. This corresponds to a fast or large movement in the hammer for a slow or small movement in the action lever. This allows the hammer to gain a high speed before striking the strings while the key only moves about a centimeter.
For the four-bar linkage to model the hammer movement, the contact point between the top of the push rod and the hammer body must be idealized as an axle pivot, joint 3 in the diagram. To match the diagram, the action would be on its side so that the hammer extends to the left joined at a right angle to r2.
- Mechanism name: Class 2 Lever
- Purpose: To apply greater force to the mutes and hammer bar than the player applies at the pedal end.
- Implementation: The pedal has a long arm that protrudes from the piano and a pull-rod closer to the axle that operates the mechanism that moves the mutes and the hammers for the sustain and una corda respectively. This allows the player to counter the force of all 88 springs for the mutes with ease.
- Governing equations:
Since de, the length of the pedal, is much greater than dl, the length from the axle to the pull rod (load), Fe can be much less than Fl. Therefore, the force applied by the foot can overcome the force of 88 springs that hold the mutes or the weight of all the hammers and the hammer bar.
Based on our analysis of the piano so far, and our understanding of the GSEE factors, three different revisions to the traditional piano could be made that would beneficially modify its design on the system level.
One of the issues with the piano is that it is too large and heavy. This limits its mobility, as well as places where it can be used. A solution to this would to reduce the number of keys in the piano itself. By cutting the number of keys in half, this would greatly reduce the size of the piano and its overall weight. This would increase its overall mobility, and its smaller footprint would allow the piano to be placed in more locations. Taking note of societal factors, the keys that would be eliminated would be those at the two extremes of the note range. This would make sure that the piano could still play a large number of piano compositions, that hover around the notes near middle C.
A second possible revision that would reduce the vertical profile of the piano at the expense of weight would be to divide the current frame system into two or perhaps three parts, and stack them on top of each other. This compact width would allow the piano frame to be oriented in a plane parallel to the keys. The piano can still be played with one action assembly because of the variation in string height. This reconfiguration would allow for a desktop version of piano.
Another issue with the full size piano is that it is difficult for children and people with small hands to extend their fingers far enough to consistently play certain chords. The solution for this would be to integrate auxiliary keys to the original piano set up, that would hit the notes of the chord with a single depressed key. This takes into account societal factors by increasing the number of people that can learn to play the piano, even before they have grown to the ideal playing size, or if they never will. This modification would also allow for greater musical experimentation, as a greater number of notes can now be played simultaneously.