Request For a Set of Eyes on an Oscillating Steel Cantilever

[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

 

[Dr.D]

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.