Request For a Set of Eyes on an Oscillating Steel Cantilever

• Jesse Millwood
In summary, a student is seeking help with modeling a vibrating cantilever with a mass at the end, as part of a project to model a Wurlitzer 200 Electric Piano. They have encountered difficulties with calculating the vertical displacement at a particular x value at a given time, and are looking for guidance on their equations. The student has also noticed that the oscillations do not die out as expected and is seeking assistance in identifying any errors in their calculations.
Jesse Millwood
Hello,
I am an electrical engineering student and I was hoping some body here could help me out with a cantilever question.

I want to model a vibrating cantilever with a mass at the end. I am doing this for a project where I wanted to model a Wurlitzer 200 Electric Piano. The way they produce the oscillations is kind of neat where there is a steel reed that is grounded and when it is struck by the key mechanism it vibrates. There is then a pickup that is kind of like a comb, where the teeth of it go in between all of the reeds corresponding with different keys. The comb pickup is pulled up to around 150V by a 1Meg resistor, since there is a difference in voltage across a distance, there is capacitance and with the vibrating reed, it makes a variable capacitor. I am fine with the capacitance calculations and modeling the rest of the circuitry but I am getting kind of bogged down with some of these (relatively basic?) calculations with the beam vibrating.

My idea to model the vibrating reed as a cantilever with a point mass on the end is to treat it as a spring mass system. I (quickly) read through some texts on the Euler-Bernoulli Beam Theory but opted to model it as a spring because in Harris' Shock and Vibration Handbook there didn't seem to be that big of a difference between the Rayleigh method and others when compared in chapter 7. If I have time at the end I will go back and get more complicated but for now I would like to just model the beam in a simple manner. I haven't had a mechanical class that dealt with cantilevers in a while and I looked at some old notes but we never covered vibrating structures, as it was a statics class. So I was thinking that I could model the cantilever as a spring mass system.

What I have so far for an example calculation is something like this:
^ y
| ///|________
| ///|________|* <-Point mass
| ///|
|-----------------------> x
Dimensions of the beam (this is for one of the F# reeds):
Length : ##36.83 mm##
Thickness : ##0.64 mm##
Width : ##3.83 mm##

Steel Properties:
Density (##\rho##) : ##8050 g/m3##
Youngs' Modulus (##Y##) : ##210e9 GPa##
2nd Moment of Inertia (##I##) : ##\frac{width \cdot thickness^3}{12} = 83.67e-15##
Spring Constant (##k##) : ## \frac{3\cdot Y \cdot I}{L^3} = 1055##
Damping Ratio (##\zeta##) : ##\frac{\pi}{L} \cdot \sqrt{\frac{1}{k\cdot \rho}} = 29.27e-3##

Cantilever (reed) Properties:
Initial Displacement (##\delta_0##): ##2mm##
mass at the end(##m##): ##1.33g##

I want to be able to calculate the vertical displacement at a particular x value along the beam at a particular time value. I am doing it this way:
##y(x,t) = \delta_0 \cdot e^{\frac{-\zeta}{m\cdot t}}\cdot sin\left(\pi \frac{x}{l} \right) sin\left( \omega t\right)##
where ##\omega =\sqrt{\frac{k}{m+0.23m}}##

I have pieced together equations from various texts that I can find for free and other websites so my main question here is :
Does anyone see any glaring inconsistencies or anything that is very wrong here?

My reason for asking for a second set of mechanically inclined eyes is when I model it this way, the oscillation does not seem to die out as I would expect. I don't know why I expect it to die out sooner but I just have a feeling that something may be miscalculated.

As it stands my simple python script plots this oscillation at 174Hz (which is what I want) but the oscillations seem to die out around 4 minutes (the plot is attached). Also When I change my ##x## value to be half of the length the oscillation at the shorter distance down the beam has a higher amplitude, I feel that it is from the ##sin\left(\pi \frac{x}{l}\right)## term in my ##y(x,t)## equation and now I can not find the source of that equation.

Thank you for any help, guidance or a friendly point in the right direction/material,
Jesse

Attachments

• figure_1.png
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Jesse, the sin(pi*x/L) factor that you have in your solution will give the solution an identically zero value at x = L. This is not what you want for a cantilever; the free end should show maximum displacement.

If this came out of a Rayleigh solution assumed mode shape, you were presumably looking at a pinned-pinned (or simply supported) beam, not a cantilever.

What is a Request For a Set of Eyes on an Oscillating Steel Cantilever?

A Request For a Set of Eyes on an Oscillating Steel Cantilever is a formal notice or invitation for other scientists or experts to review and analyze a specific steel cantilever structure that exhibits oscillating behavior. This request is usually made when the original researcher or team is seeking additional insights, expertise, or validation for their findings.

What is the purpose of a Request For a Set of Eyes on an Oscillating Steel Cantilever?

The purpose of a Request For a Set of Eyes on an Oscillating Steel Cantilever is to gain a fresh perspective and potentially identify any errors, flaws, or limitations in the original research or analysis. This can lead to a more comprehensive understanding of the behavior of the steel cantilever and potentially improve its design, construction, or application.

Who can make a Request For a Set of Eyes on an Oscillating Steel Cantilever?

Anyone who has conducted research or analysis on an oscillating steel cantilever can make a Request For a Set of Eyes. This can include individual scientists, research teams, or organizations. The request can also be made by a third party who has a vested interest in the steel cantilever, such as an engineering firm or government agency.

What is the process for responding to a Request For a Set of Eyes on an Oscillating Steel Cantilever?

The process for responding to a Request For a Set of Eyes on an Oscillating Steel Cantilever may vary depending on the specific circumstances. Generally, the responding scientist or expert will review the original research and provide feedback or analysis based on their own expertise. This can include identifying any errors or limitations, proposing alternative approaches, or validating the original findings.

What are the potential benefits of responding to a Request For a Set of Eyes on an Oscillating Steel Cantilever?

Responding to a Request For a Set of Eyes on an Oscillating Steel Cantilever can have several benefits. It can provide the original researcher or team with valuable feedback and insights, potentially leading to improvements in their work. It can also expand the scientific community's understanding of the behavior of steel cantilevers and contribute to the development of more robust and safe structures. Additionally, responding to a request can also increase collaboration and networking opportunities among scientists and experts in the field.

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