# Coupled damped harmonic oscillator

Hi everyone,

I'm dealing with system identification for the first time in my life and am in desperate need of help :) The system is spring-mounted and I'm analyzing the vertical and torsional displacements. However, it seems like the vertical and torsional oscillations are coupled (shouldn't be in theory..). So I want to create a model that takes this coupling into account. However this turned out to be much harder than I first thought...

The classical damped harmonic oscillator
x" + 2*gamma*x' + omega_0^2*x = 0 (1)

can be rewritten as:
x(t) = A*exp(-gamma*t)*cos(omega*t + phi) (2)
if omega_0^2 > gamma^2, (omega=sqrt(omega_0^2-gamma^2))

I'm supposing a coupled system between vertical and torsional oscillations would look something like this:
h" + 2*gamma*h' + omega_0^2*h + k1*a' + k2*a = 0 (3)
a" + 2*gamma*a' + omega_0^2*a + k3*h' + k4*h = 0 (4)

where k1, k2, k3 and k4 are coupling coeff.

But from here I have no idea how to continue... The main goal is of course to determine the coupling coeff and validating the model with experiment data. Does anyone have any suggestions of how to determine the coupling coeff? Or know how to express (3) and (4) as h(t) and a(t)? Furthermore I'm not sure that eqs (3) and (4) are the best to describe this coupled system... Any suggestions at this point are highly appreciated!

Thanks,
Maria

What type of spring configuration is responsible for the motion? It seems to me you either need a model based on the geometry and properties of the actual device, or you need a statistical/numerical model that can extract a coupling factor from measured data.

The system is a sectional bridge deck model, similar to the one found here (figure 3):

I want to create a model based on measured data but with parameters that have physical meaning.

I've done vertical and torsional step responses. From the frequency domains it is easy to detect the natural frequencies but there are some other disturbing peeks that should not be there... Some of them I consider to be noise, but one (at least) is to big to be noise and I want to find some sort of physical meaning to it... To begin this investigation I wanted to see if there is coupling between the torsional and vertical oscillations.

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My background is EE but I've been doing some basic simulator work. It looks like there are four helical springs above and four below the deck.

First question, does a helical spring inject a torque on when it compresses? My intuition says it does unless there is decoupling provided at the joint of the spring with the deck. So in terms of physical models I'd look into that, then run a simulator with a simplified system to get some insight.

Sorry if I'm missing your point ...

Also another thought ... if the actual spring rates are poorly matched I wonder if the vertical displacements might then couple torsional oscilations, or something to that effect?

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I think you have a point about the spring configuration causing disturbances in the frequency domain (I hope this was your point...). I've plans to improve the physical model by installing a joint where the springs are attached. I'm hoping that this will lessen or remove some peaks.

However, if there still are unignorable peaks left I might want to expand the model so that it includes them. Since my first post I have though realized that the equations I proposed for coupling are only capable of producing peaks at the natural frequencies (the disturbing ones I have are not at the natural frequency or a multiple of it...).

So what I would want to know (even if it proves to be unnecessary after my model improvements, it would still be interesting) is how, if there is some general equation for it, to describe this type of "coupled" system. With "coupled" I mean that there are frequency peaks other than the natural ones that are repeated in both the vertical and torsional the frequency domains. This shows that there is a clear dependence, but how to describe it?

If time and budget permit, I would first create a rig to measure the coupling or lack thereof of torque with compression or elongation of a single spring, using a rig that permits testing of different joint methods to the deck.

Next consider whether an imbalance in the linear spring characteristics or torque coupling characteristics creates an oscillation at some frequency (probably using the model for a single spring/joint combination in a computer simulator for rapid feedback).

In an electrical circuit analog springs are like inductors, mass is like a capacitor, and damping is like an lossy conductor, and when you put these elements in series/parallel configurations, the imbalances between actual device values become significant. So I'm assuming parameter imbalance may be causing a frequency in your noise signal ...

Personally my math skills would not permit a differential equation solution for eight springs in series/parallel with so many degrees of freedom and torsional coupling ... I'd end up chasing my tail in the math. DYNAST shell is one tool I use but I use it for much simpler systems than yours.

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Another question, doesn't the flow of air past an object cause oscillations that could account for your results? For example, an open jacket on a motorcycle rider flaps up and down on each side at a natural frequency due to alternations in shedding a vortex. The back end of a big rig has a side to side (and perhaps torsional coupling) as the vortex sheds on one side, then the other, in a cyclic process, if I'm not mistaken. Hope that is useful as I don't know your exact problem ...

http://en.wikipedia.org/wiki/Vortex