Primary Question: How can you determine the spring constant $k$ of an elastic material?
I was recently tasked with finding the spring constant for a series of elastomeric materials, the first of which was a silicone rubber.
My Approach
In an attempt to determine the spring constant with limited resources I used a known mass which I attached to my spring using some clamps that I $3D$ printed. I made a simple backdrop with some paper and a ruler and recorded myself dropping the weight using a camera.
Using a simple motion capture program (Tracker) I was able to very precisely plot the motion of the mass as it oscillated.
I then plotted that data and applied a non-linear curve fitting method to the discrete dataset so that I could then characterize the motion and calculate the spring constant.
Knowing the data was that of an underdamped harmonic oscillator I applied the following equation from the source listed below.
[credit] http://web.mit.edu/8.01t/www/materials/modules/chapter23.pdf
$$ x(t) = x_{m}e^{−αt} cos(γ t +φ) \quad\quad\quad (1) $$ Where, $x_{m}$ is a constant related to the amplitude of the underdamped oscillation, $α$ is a parameter related to the rate of decay, $φ$ is the phase constant, $b$ is the constant of proportionality, $m$ is the mass, and $γ$ is the angular frequency of oscillation of the underdamped oscillator.
Note: $$ x_{m} = \sqrt{C^2+D^2} \quad\quad\quad (1a) $$ $$ φ = arctan(-\frac{D}{C}) \quad\quad\quad (1b) $$ $$ \frac{d^2x}{dt^2}+\frac{b}{m}\frac{dx}{dt}+\frac{k}{m}x = 0 \quad\quad\quad (1c) $$ C and D are arbitrary constants that result from the solution to the differential equation (1c)
$$ γ = \sqrt{(\frac{k}{m} − (\frac{b}{2m})^{2})} \quad\quad\quad (2) $$ $$ α = \frac{b}{2m} \quad\quad\quad (3) $$
If we substitute (3) into (2) and square both sides we get the following, $$ γ^{2} = \pm((\frac{k}{m} − \alpha^{2}) \quad\quad\quad (4) $$ After a little algebraic manipulation we arrive at $$ k = m(\alpha^{2}+\gamma^{2}) \quad\quad\quad (5a) $$ $$ k = m(\alpha^{2}-\gamma^{2}) \quad\quad\quad (5b) $$
knowing that all of the terms that make up the equation for $\gamma$; k, m, and b are all positive constants we can use (5a) and disregard (5b).
$$ k = m(\alpha^{2} +\gamma^{2}) \quad\quad\quad (6) $$
After applying the non-linear best fit approximation I had all of the constants that I needed to solve for k.
Because $k$ is calculated in terms of mass $m$ I assumed that regardless of the mass used in the test the value of $k$ would remain the same and therefore testing the same material using different masses would serve as a confirmation of the spring constant calculated. This, however, was not the case.
The larger the mass used the lower the value $k$ was calculated to be.
Secondary Question: Are any of the assumptions used here incorrect?
- If a dataset clearly indicates it has an underdamped harmonic motion is there any reason not to use the equations stated above to characterize such data?
- Is it acceptable to make the assumption that $\gamma$ will never be negative?
- Is it wrong to assume that the value of $k$ would be constant for an elastomer?
