Group 1 2012 Gate 3

The purpose of this gate was to take our dissected piano and complete an in-depth analysis of its components and subsystems. The first step towards this goal was to document all of the parts and summarize their function, material, and manufacturing process. The geometry, mass, and properties are listed in the Component Summary. We then considered the factors that affected these properties. Several components were detailed in the Global, Societal, Economic, Environmental model (GSEE) in relation to their material, geometry, and manufacturing processes. Due to the complex interactions within the action, several key components in the action and hammer systems were modeled in a CAD software package. We proposed a method for improving the strings through an engineering analysis. This includes a discussion of testing materials and a mathematical analysis of the materials. We listed several other possible design revisions that reduce cost, improve quality, or ease manufacturing.

Project Management

Cause for Corrective Action

The group seems to be working together well now. We generally have very regular attendance at meetings and are improving our focus during those meetings. The cause for concern now is more refining our approach to group assignments that they may be completed as quickly and efficiently as possible. To that end, everyone was specifically assigned tasks before our first meeting in an attempt to streamline group work, and also to attempt to more evenly distribute out of meeting work among members.

Project Evaluation

Component Summary

Component Name	Component Picture	Material(s)	Manufacturing Processes	Number of Components	Description
Keys		Wood, with plastic covering	Wood parts sawed, Plastic adhered	88	Pressed by user in order to trigger action to hit desired string.
Hammers		Wood body, head is felt	Turned and sawed wood body, Needle Stitched Felt, parts then adhered	88	Strike the string in order to produce a note. (Red box in picture)
Pull rod		Steel, wooden end	Drawn Steel, Milled Wood, Adhered together	88	Transmits key energy to the action assembly.
Action Assembly		Wood, Steel	Sawed Wood, Several Drawn Steel Pins, Drawn and rolled sring	88	Transmits pull rod energy to the hammers, the internal spring resets the hammer to its rest position. (Yellow box in picture)
Key rack		Wood, Steel, Felt	Sawed wood, Drawn Steel, Needle Stitched Felt, adhered together	1	Holds the keys in place.
Hammer rest rack	Need Picture	Wood, fabric	Sawed wood, fabric adhered	1	Holds the hammers in the proper position while they are resting, moves forward when una corda pedal is pressed.
Mutes		Wood, steel, felt	Milled and Sawed Wood, Needle Stitched Felt, Drawn Steel	88	Rest on strings to dampen the sound produced when the strings are struck.
Pedal Assembly		Metal (most likely bronze or copper), Wood, Felt, Steel pins and screws	Cast Bronze, Drawn Bronze, Needle Stitched Felt, Sawed Wood	2	There are two pedals, the mute and the una corda, which respectively move the mutes off the strings and the hammers closer to the strings. The pedal assemblies hold the pedals in place so the user can depress them, provide a pivot for the pedal itself, and translate the pedal motion to the pedal bars by use of a long straight rod.
Pedal Bars		Wood, Steel	Sawed Wood, Rolled Steel	2	Transfers kinetic energy of the pedals to the pedal dowels, returns to place via the stored energy of the flat spring connecting it to the piano frame floor.
Piano Casing		Wood	Sawed Wood, joined with screws and or adhered together.	1	Outer skeleton of piano, provides support and protection to all other components.
Frame		Cast Iron	Cast Iron	1	Heavy iron body that resists string tension.
Action Body		Wood	Sawed Wood	1	Provides support for all the components of the action.
Mute rod		metal	Cast iron	88	Connects the mute to the rest of the piano.
Wheels		Steel, Hard plastic	Rolled Steel, Injection Molded Plastic	4	Make the piano easier to move.
Strings		Steel	Drawn Steel	233	Resonate at certain frequencies when struck. Larger strings produce lower frequencies. The dustless spots on the picture are where the mutes rest. (Click on image at left to see all strings at once)

Product Analysis

Component Complexity

The majority of the components of the piano are rather simple; most are just portions of levers or linkages. That said, there is a rough correlation between the number of fixed parts on a component and its complexity. We define a complexity scale such that:

1. Entire component is one solid body of one material.
2. Entire component is one solid body composed of multiple different materials.
3. Component is composed of one solid body with permanently attached moving parts.

Keys

The keys have only one function: to initiate the transmission of kinetic energy and signal from the user to the piano action. They are one of the few moving components on the outside of the piano.

Component Form

Shape	Primary Dimension	Component Complexity	Dimensions	Weight	Material	Surface Finish
Rectangular Prism	2 Dimensional	1	13 1/2 x 7/8 x 9/8 in	200-300g	Wood	Smooth plastic covering

Component Form Analysis

As the primary function of the keys is to transmit kinetic energy and signal through lever action, the keys relatively large length as compared to width allow them to require relatively little force to fully strike the key.

The key is made of wood and plastic primarily for economic reasons; alternative key materials (notably ivory) exist, but wood and plastic are the least expensive and most widely available.

The plastic coating is largely an aesthetic consideration; the keys do not need them to function, but they mimic the traditional key coloring of older and more expensive pianos. In addition, the black and white colors serve to distinguish keys for the user.

Manufacturing Methods

The keys are made of sawed wood with plastic laminated on top of them. The long and generally linear shape of the keys lend themselves to sawing.

Strings

The function of the strings is to produce the sound that is desired from acoustic energy and acoustic signals.

Component Form

Shape	Primary Dimension	Component Complexity	Dimensions	Weight	Material	Surface Finish
Linear	1 Dimensional	1	7-40 in long 1/16-1/4 in diameter	Unknown	High Carbon Steel	Ribbed

Component Form Analysis

The strings are the component of the piano that actually produce the sound. There are 88 strings of various lengths inside the piano to produce the different sounds.

The strings are made of high carbon steel because they are strong enough to be held in the high tensions required to produce the sounds. The same strings are used today for the societal reason of the recognizable sound of a hammer striking a string that is associated with a piano.

The strings have a ribbed surface since they are kept in wound in high tension, making them no longer smooth.

Manufacturing Methods

The strings were made using a drawing process. Steel is a material that can be easily drawn into wire, and its is relatively easy to achieve the different lengths and thicknesses of the strings using this process.

Hammers

The function of the hammers is to physically hit the strings with the amount of force applied on the keys

Component Form

Shape	Primary Dimension	Component Complexity	Dimensions	Weight	Material	Surface Finish
L-Shape	2 Dimensional	3	2 1/2 x 6 x 3/8 in	200 g	Wood/Metal/Fabric	Smooth Wood

Component Form Analysis

The hammer is the component in the piano that uses kinetic energy to strike the strings. There are 88 hammers total, with one designated for each string. The hammers are made of wood to ease their functionality, being light for quick accurate strikes. They have soft fabric on the end to soften the sound when hitting the string. Metal wires connect the hammers to the action. The surface finish is not important on the hammers since they are not designed to be touched or seen.

Manufacturing Methods

The hammers were sawed from wood and sanded down to make the round features.

Mute

The function of the mute is to dampen the sound coming off of the string.

Component Form

Shape	Primary Dimension	Component Complexity	Dimensions	Weight	Material	Surface Finish
Bent Linear	2 Dimensional	3	7 1/4 x 3/8 x 7/8 in	100 g	Wood/Metal/Fabric	Smooth Wood

Component Form Analysis

The mute acts to dampen the sound coming off of the string. Its normal position is touching the strings, but it can be moved off of the string by hitting the pedal with your foot. The mute is made of mostly wood pieces with some metal portions. The end of the mute itself is a soft fabric as this is the part that is in direct contact with the string. Aesthetics are not an issue for this component as it is never intended to be seen, but it does have a smooth sanded wood finish.

Manufacturing Methods

The wooden pieces of the mute were made using a saw. This is apparent due to the material and the small complex cuts that could be done by hand.

Pedals

The function of the pedals is to allow the user to augment the sound using their feet.

Component Form

Shape	Primary Dimension	Component Complexity	Dimensions	Weight	Material	Surface Finish
Rounded Rectangular	3 Dimensional	1	2 x 1/2 x 7 5/8 in	1000g	Cast Iron	Smooth and Shiny

Component Form Analysis

The pedal is depressed by the user in order to initiate the movement of the mutes off of the string to change the output sound. The pedal is made of cast iron mostly due to the strength and wear that it has to withstand from the user pressing it forcefully with their feet. On the surface the pedal is a shiny gold-bronze color, because it is one of the few components that is visible from the outside at all times.

Manufacturing Methods

The pedal was manufactured using a cast because its shape is irregular, and a cast could be easily filled with iron to produce that shape with the strength needed for its application.

Frame

The function of the frame is to hold all of the components of the piano together and protect them from the outside.

Component Form

Shape	Primary Dimension	Component Complexity	Dimensions	Weight	Material	Surface Finish
Rectangular	3 Dimensional	2	54 1/2 x 24 1/2 x 34 in	25 kg	Wood	Smooth Stain

Component Form Analysis

The frame of the piano is the largest component. It is responsible for maintaining structural integrity of all the interior components, as well as protecting them from the outside. Wood was used to build the frame because it is structurally sound, relatively cheap, and readily available. On the surface the frame has a dark wood stain that is smooth to the touch. This is because this part of the piano is always seen whether it is or is not in use.

Manufacturing Methods

Because the frame is made of wood, all of its parts could be easily manufactured with simple wood cutting saw processes, except for the legs that would require more complex carving due to their curved shape.

Action End Brackets

The function of the action end brackets is to hold all of the parts of the action together and attach to the frame.

Component Form

Shape	Primary Dimension	Component Complexity	Dimensions	Weight	Material	Surface Finish
Irregular Polygon	2 Dimensional	1	12 1/2 x 3 x 1/2	500g	Cast Iron	Coarse, Rough

Component Form Analysis

The purpose of the action end brackets is to hold the parts of the action together structurally, as well as attach to the frame. These end brackets are made of cast iron to ensure strength and its irregular shape is due to the odd shape of the action itself and the locations of the connections on the action and the frame. It surface is coarse and rough because the aesthetics of this component are irrelevant to the design as it is an internal part of the piano that is never seen, so no fine machining or shine is required.

Manufacturing Methods

Since the end brackets have such an irregular shape and they need to be strong, these components were made by casting iron.

Solid Modeled Assembly

We decided to model both the hammer and pull rod subsystems in order to better visualize the structure and basic interaction between the subsystems. Autodesk Inventor Professional 2010 was used to model the components, due to members having multiple years of experience with the program.

Hammer assembly and exploded assembly views:

Hammer components:

Pull rod subsystem assembly and exploded assembly views:

Pull rod subsystem components:

Pull rod and hammer subsystem orientation:

Engineering Analysis

Overview

The function of converting mechanical energy to sound energy depends on many variables ranging from material to internal stresses to relative orientation to other parts. Consequently, the strings and soundboard would be the best candidates for an engineering analysis.

The existing strings are predominantly steel, although the lower-pitched strings have a copper wrapping to increase their unit weight per length.

The known problems that affect the strings in all pianos are issues of harmonics. A single string can produce several simultaneous standing waves at multiple frequencies according to the harmonic series. Since the scale system of western music is based on this series, it is important that each harmonic is properly tuned. However, at each overtone, errors develop because the stiffness of the string has more effect than the tension. This is a phenomenon known as inharmonicity and it is the topic of the strings' engineering analysis.

The overall goal of the engineering analysis is to create a string that responds consistently to all frequencies and produces as close to perfect overtones following the harmonic series. This would allow a constant sound across the range of the piano.

Equipment and Materials

To analyze the strings, it is necessary to measure the multiple frequencies and amplitudes of the sound waveform generated by the string. The best equipment for this is an oscilloscope connected to a microphone. To isolate various harmonics, it may also be helpful to connect a variable band-pass filter to the audio signal line. A band pass filter blocks out any frequencies below a set lower limit or above a set higher limit in an audio signal. Since the multiple frequencies of harmonics present in a piano note each need to be measured, a band-pass filter would prove very useful.

The procedure for analyzing the strings needs to cover the range of pitches on the piano and a variety of materials. Also, since existing pianos use a wrap of copper on the steel base wire, the analysis must include the possibility of multiple materials and variable cross-sections.

Since the tension is a key factor, each material's yield strength and ultimate tensile strength need to be measured. The string should be attached to a scale and tensioned until deformation or breaking. The maximum safe tension for the remainder of the testing should allow an adequate safety margin below the yield strength.

To test each string, the string must be tensioned on a rigid structure. However, since length affects pitch, the testing structure must be adjustable in length and have a traditional tuning peg to tension the wire precisely. The test stand must also include a hammer, action, and key to strike the string as in a real piano.

The complete equipment list is as follows:

• Oscilloscope
• Microphone
• Adjustable audio band-pass filter
• Scale
• Testing stand with the following features:
  • Adjustable length
  • Tuning peg style adjustable tensioner
  • key-action-hammer mechanism

The strings for testing must have variable material, thickness, and length. Suitable strings may include but are not limited to common piano wire, high-tension steel wire (multi-strand or single-strand), aluminum, copper, and plastics. Of course, if, during the testing, trends emerge that suggest other materials, those materials may also be tested.

Each material should have multiple cross-sectional areas and lengths available. Combinations of the materials (such as the current copper wrap-steel core combination in our piano) could also be candidates for testing.

Procedure

The following steps detail the procedure for one string type. Repeat these steps as necessary until all strings have been tested. Each string should be tested from a length and tension that yields approximately 27.5 Hz (the lowest note on piano) to 4186 Hz (the highest note) in increments of roughly double the frequency (an octave on the piano) [1]. Use (eq. 1) in Analysis to predict the frequency.

The following data table can be used to gather all the necessary data.

Data Table:

Trial

Material

Diameter

Length

Max Tension

Actual Tension

λ₀

f₀

λ₁

f₁

λ₂

f₂

…

λ_n

f_n

String 1, octave 1

String 1, octave 2

String 1, octave 3

…

String 2, octave 1

…

String n, octave n

1. Place string in tension mount with scale and apply tension until deformation or failure. Record the maximum allowable tension.
2. Based on the maximum tension in the previous step, place the string in the testing stand. Leave a factor of safety of 3.0 to 4.0.
3. Record the length and tension in the string.
4. Strike the hammer and observe the trace on the oscilloscope.
5. Adjust the band-pass filter to allow only the lowest frequency appears on the oscilloscope. This is the first harmonic, f₀.
6. Measure the period and amplitude of the trace on the oscilloscope. The period corresponds to a time in seconds depending on the scale and the amplitude corresponds to the voltage of the electrical audio signal. Calculate the frequency using the formula f=v/λ where v=speed of sound in air=340m/s and λ=period (s).
7. Adjust the band-pass filter until the next lowest frequency shows on the oscilloscope.
8. Measure the period and amplitude of the trace on the oscilloscope. Calculate the frequency using the formula f=v/λ where v=speed of sound in air=340m/s and λ=period (s).
9. Repeat steps 7 and 8 until the overtone frequency is above 20,000Hz. Above 20,000 Hz, humans cannot detect sound.

Analysis

The ideal frequency of the first harmonic is given by the formula:

(eq. 1) [2]

Where f₀=first harmonic frequency, T=tension, ρ=density, S=cross-sectional area, and l=free string length. This formula assumes that the string has negligible stiffness.

The frequency of the nth harmonic of an ideal string is given by:

(eq. 2) [3]

Where n is a natural number. The ideal nth harmonic will have a frequency n-times that of the fundamental frequency.

To find the optimum material, the string must follow the harmonic equation (eq. 2). The data should be entered in a spreadsheet program. The ideal harmonic frequencies should be compared to the actual frequencies for each string. The ratio can be averaged for each string. The material, cross-section, and tension that yield the ratio closest to 1 is the ideal string.

Conclusions and Further Action

Once the best material, cross-section, and tension to reduce inharmonicity are found, all the strings should be changed to match. However, since the strings were tested at multiple octaves, there may be a different optimum material at different frequency ranges like our current piano. Like the lowest notes use a thicker, composite wire made of steel and copper in the current piano, a certain string may be best in one octave but not another. If the ratio is closer to 1 for different materials in different octaves, the string may change properties across the range of the piano.

Design Revisions

Casters

The casters the piano initially came with were both small and poorly secured into the wooden frame of the piano. One had fallen out before we purchased the piano, and another fell out as we were transporting it. We propose that the casters be replaced with significantly larger versions and that they be secured with an actual bracket instead of a press-fit pivot in a drilled hole. This is largely an environmental factor improvement, as the larger, more robustly secured wheels would both help improve overall product lifespan and enable it move more easily over more varied or difficult terrain.

Soundboard

Our soundboard is composed of laminated planks of what is most likely spruce to improve sound projection of the piano. Spruce is generally considered to be acoustically the best material available, however, it is rather expensive. At the expense of sound quality, the cost of the piano could be decreased by replacing the material used in the soundboard with something cheaper and easier to produce, like plywood This would make the piano available to a wider societal demographic.

Hammer and Action Bodies

Our piano has 88 sets of hammer and action bodies, all composed of glued together segments of drilled out wood. It would be much cheaper and faster to form all of those pieces out of plastic using injection molding. The economic benefits of this could then be transferred to the consumer.