2-DoF system characterization with 2 dual axis accelerometers

[Cmdr_Ryder]

I've been having some trouble with this problem for a while. (Its not a homework or coursework problem)
Its a part of an experimental set up that I'm building and this is a simplified version of it.

I have a rigid rod which is hinged at a point and can pitch or heave about that location. There are 2 dual axis accelerometers mounted at either ends with the one at Leading edge inclined at a known angle theta.
I need to calculate the real time angular and vertical displacements from the accelerometer outputs. The angle is the most important parameter that I need.
The angles are not small, so small angle assumptions does not work.
The motion of the system is quite random, and its a non-inertial frame of reference. (This system is quite similar to the suspension system modelling for cars)

I'd be grateful for any help or insight regarding the problem.
Thanks

 

[.Scott]

I take it that the hinge is at that "x" point in the circle.
I am not clear on the function of that hinge.
Are all displacement forces applied through that point?

Obviously, if this is in free fall, all four inputs are zero and there is no solution.
I believe that you are taking the negative y-axis as down. And that you are presuming that there is a persistent gravitational pull being reported by the accelerometer outputs. You are going to need to rely on the fact that the gravity is the only persistent force.

So you need to know over what period of time will other forces tend towards zero.

The fact that one of the accelerometers is tilted by omega is fine.

1) Start by changing the raw input from A1x and A1y to the same coordinate system as A2.
2) Then subtract those vectors (A1 and A2). This will yield a vector that will reveal the rotation rate and and a translational acceleration.
3) The rotation rate will cause the gravitational force to show up as a rotating force. Synthesize two sine waves based on the rotation rate, each 90 degrees offset from the other. This "sine wave" will vary in frequency as the rotation rate changes.
4) Multiply each of those sine waves with the translational force and average those over a long enough period to filter out non-gravitational influences.
5) The resulting vector, with an angle relative to the sine waves you synthesized, will direct you down.

At this point you can do two things:
1) Go back and compute the angles from the start of the test.
2) Continue to track those angle through the changing rotation rate so that you can generate angles in real time.

Now, if this is a land-based vehicle, not a drone, we can make some assumptions about how radically that angle can change relative to the horizon.
Then you could take advantage of that to determine the rod angle relative to down more precisely and quickly.

 

[Cmdr_Ryder]

Hi Scott

In real experiment,the rod is an airfoil that is in Freeplay pitch and heave motion. The hinge point is X and the rotation is along that point. The forces on the airfoil are non uniform and kinda random.
The whole airfoil would be oscillating along the hinge point. It's a 2 dof system with pitch and heave.
I did derive a set of equations using geometry, but it becomes complex 2nd order ode if I don't use the small angle assumption.

Regarding your answer, I didn't quite understand the sine wave part.
I must admit, mechanics is not really my field and so,I might be a bit slow in grasping it.

 

[.Scott]

Basically, you are setting up an arbitrary Cartesian coordinate system - with an initially unknown relation to vertical and horizontal.
Then you are keeping track of the non-rotational forces based on that coordinate system. I called it a sine wave simply because I was presuming continuous rotation - and so you would be multiplying these two "sine waves" with your non-rotational forces to get the forces in the X,Y of your coordinate system.

But if you are not rotating continuously, the angle will be moving this way and that and you will be multiplying by the sine and cosine of an angle that is far from a monotonic function of time. In that case, you are going to have to look at both the angular acceleration and the centrifugal force very closely to deduce the change in angle relative to you coordinate system.

So, from the 4 measurements, you get these:
The difference in acceleration relative to the length of the rod: this is your centrifugal force.
The difference in acceleration perpendicular to the rod: this is what changes your angular momentum and eventually tells you which way the rod is turning.
The average acceleration X,Y: This is what you convert (multiplying by sine and cosine of the deduced change in angle) to you reference coordinate system.

 

[Cmdr_Ryder]

Basically, you are setting up an arbitrary Cartesian coordinate system - with an initially unknown relation to vertical and horizontal.
Then you are keeping track of the non-rotational forces based on that coordinate system. I called it a sine wave simply because I was presuming continuous rotation - and so you would be multiplying these two "sine waves" with your non-rotational forces to get the forces in the X,Y of your coordinate system.

But if you are not rotating continuously, the angle will be moving this way and that and you will be multiplying by the sine and cosine of an angle that is far from a monotonic function of time. In that case, you are going to have to look at both the angular acceleration and the centrifugal force very closely to deduce the change in angle relative to you coordinate system.

So, from the 4 measurements, you get these:
The difference in acceleration relative to the length of the rod: this is your centrifugal force.
The difference in acceleration perpendicular to the rod: this is what changes your angular momentum and eventually tells you which way the rod is turning.
The average acceleration X,Y: This is what you convert (multiplying by sine and cosine of the deduced change in angle) to you reference coordinate system.

Hi Scott

Thanks for the help. I think that method works. Let me add it to my code and test it.

 

[Cmdr_Ryder]

Hi Scott

Thanks for the help. I think that method works. Let me add it to my code and test it.

If we consider the differential vertical accelerations as a function of angular momentum, aren't we neglecting the vertical motion of the rod?

I tried using force balances to generate the equations, but I'm not sure it works because the whole system is non inertial. I'm quite confused, as this problem seems very similar to bicycle model for car suspensions. And I'm always ending up at a force and torque balance equation.

Using small angle assumption, the equation that I got is

a2 = -x2 A + Y
a1 = x1 A +Y sin(theta)

Where A is angular acceleration and Y is vertical acceleration (a1 and a2 are resultants)