Investigation Results of Damping of A Spring Showing Changing Phase Angle? Why?

Question

In an experiment I've recorded the displacement of the spring over time, investigating underdamped simple harmonic motion.

Using pre-existing formulae the data should conform to a curve of the form

$$x(t)=Ae^{-\gamma\omega_0t}\cos(t\omega_d-\phi)$$

where

$$\omega_d=\omega_0\sqrt{1-\gamma^2}$$

and

$$\gamma=\frac{c\text{ (damping constant)}}{2mk}.$$

However applying this to collected data results in eventual offset,

Applying the above equation it accurately models the data intially,

But as time goes on it gradually moves out of phase,

Why is this? Through tinkering with the formula I can accurately model the phase change by letting $\phi$ be a function of time, that is

$$\phi=\phi_0+Bt$$

where $B$ is some other constant. As a result of this the data is modelling perfectly but why?

I think it could have something to do with the experiment setup as shown below,

The spring is pulled down by hand so there is some small swinging, like a pendulum but I don't understand how that would translate to a linear function for $\phi$.

All help and ideas appreciated!

Answer 1

Welcome to the experimental world!

To be fair, it's quite normal that oscillations do not perfectly coincides after some periods. You have a period of the oscillation $T \simeq 0.7 \, s$ and a shift (if minimal) of $0.3 \, s$ after $300 \, s$. This means an approximate error on the period of $\Delta T = 0.3 \, s \cdot \frac{0.7 \, s}{300 \, s} = 7 \cdot 10^{-4} \, s$, i.e. a relative error of $\frac{\Delta T}{T} = 1 \cdot 10^{-3} = 0.001$, i.e. a $.1 \%$ error.

Structural dynamics usually deal with modes (shapes, frequencies and damping coefficients) of a structure, and not with time histories over long time. I could say that $.1\%$ relative difference between the natural frequency of the numerical model and the experimental model is good enough for everyone.

Quantify your error.

Example 1 - Undamped oscillator. For an undamped oscillator with mass $m$ and stiffness $k$, the natural angular frequency and the period are

$$\omega = \sqrt{\frac{k}{m}} \quad , \quad T = \frac{2 \pi}{\omega} = 2\pi \sqrt{\frac{m}{k}} . $$

Let's assume here for simplicity that the only source of errors are different values in $m$ and $k$, and no other factors not included in the model. The error on $\omega$ due to a small difference in the value of $m$ and $k$, with the sensitivity analysis, reads

$$ \Delta \omega = \frac{\partial \omega}{\partial k} \Delta k + \frac{\partial \omega}{\partial m} \Delta m = \frac{1}{2}\frac{1}{\sqrt{km}} \Delta k - \frac{1}{2} \sqrt{\frac{k}{m^3}} \Delta m $$ $$ \Delta T = \frac{\partial T}{\partial k} \Delta k + \frac{\partial T}{\partial m} \Delta m = -\pi\sqrt{\frac{m}{k^3}} \Delta k + \pi \sqrt{\frac{1}{km}} \Delta m $$

while the relative error is obtained, dividing by $\omega$ and $T$ respectively,

$$\frac{\Delta \omega}{\omega} = \frac{1}{2} \dfrac{\Delta k}{k} - \frac{1}{2} \frac{\Delta m}{m} \ .$$

$$\frac{\Delta T}{T} = -\frac{1}{2} \dfrac{\Delta k}{k} + \frac{1}{2} \frac{\Delta m}{m} \ .$$

Thus, with perfect mass, a $+1\%$ relative error on the measure of the stiffness produces a $+0.5\%$ relative error on the angular frequency; with a perfect stiffness, a $1\%$ relative error on the mass produces a $-0.5\%$ relative error on the angular frequency.

The $.1 \%$ relative error you get on the period, it would be produced by a $.2 \%$ relative error on the mass (on the stiffness) for perfect stiffness (mass).

Example 2 - Damped oscillator. You could do the same analysis for a damped oscillator, including the damping coefficient in the analysis. Damping is usually harder to be modeled since it comes from different processes that are only included with a damping coefficient (usually a too simplistic model).

Possible sources of error

It's likely that:

• some parameters of your system don't "precisely" coincide with the values you used in the analytical model/formula;
• some features of your system are not "perfectly" represented by your model: this is a typical case for:
  • physical constraints and boundary conditions of models in structural dynamics problems, like the one of your interest;
  • damping from air, or structural damping

The meaning of world "precisely"

What do we mean by "precise"? We usually mean "precise enough" for the application/the tolerance we need. It's likely that you can't build a numerical model that gives the very same results of experimental tests, given all the possible sources of errors, and uncontrollable uncertainty. When dealing with critical applications, some safety coefficients are used.

Approach to experimental tests and tuning of numerical models

A numerical model of a physical system is usually naturally affected by modeling errors. It's a good practice to have some "tunable" parameters in the numerical model and tune them (parameter identification) to make the prediction of the numerical model match with the measurements performed during some experimental test.

Answer 2

I think this is just because there isn't going to be perfect agreement between the frequency in your fit function and the real-life frequency. Below are two sine functions that differ in frequency by 1%. For the first few cycles, they match great.

Here are the same two sine functions many cycles later

Any error whatsoever in your frequency will result in long term phase disagreement.

Answer 3

Note:

$$ \cos(\omega_dt + (\phi_0 + Bt)) = \cos(\omega't+\phi) $$

with

$$ \omega' = \omega_d + B $$

so your damped frequency is wrong.

So look at the above answer regarding error propagations, since you're doing:

$$ \omega_d^2 = \omega_0^2 - \alpha^2 $$

where $\alpha$ is some combination of $\gamma$ and $\omega_0$. How well did you measure them?

Presumably $\alpha \ll \omega$, and quadrature differences can be a little tricky.

You need to plot the predictions for $\pm \sigma$...which may catch the initial deviation, but ofc will eventually become a scattered mess.

You also want to Fourier transform $x(t)$, maybe a short-term STFT and see if the spectrum is stationary, but you should be able to get a peak around $\omega_d$ and verify the width is consistent with your error analysis.

Finally, the pendulum with a small transverse deviation is a solved problem...what does a little angular momentum do to the eigenvalues (frequencies) of the system? Sounds like a fun perturbation theory problem.