Determining the deflection of a support rod during acceleration

[rafehi]

Hi all,

I'm currently designing a housing for a camera and omnidirectional mirror for a robot. The housing is connected to the robot through a rod of about 25mm diameter.
...{...}... (housing)
...|... (rod)
...(___)... (robot base)

I'm trying to compute the deflection in the rod when the robot accelerates (or deccelerates) at its max of 1m/s^2 but not sure how I'd go about it.

Conceptually, when the robot is at rest the rod would act as a column with a concentric load. Once the robot accelerates, the inertia of the rod and housing would cause it to want to stay, hence causing the deflection. Once it begins to deflect, the housing becomes an eccentric load and causes the rod to deflect more. The dimension of the rod is fixed so I have to ensure the housing is light enough to minimise deflection and vibration.

At this stage I'm looking at doing a rough FEA simulation and seeing what numbers it spits out but how would you go about solving such a problem analytically?

 

[Mech_Engineer]

Just consider the rod to be a cantilever beam with a force at the end of it, and calculate the maximum deflection based on the acceleration force on the camera.

http://www.efunda.com/formulae/solid_mechanics/beams/casestudy_bc_cantilever.cfm

 

[rafehi]

Is it really that simple? The base of the robot is what's accelerating, not the camera (housing). I guess we can use the robot as the reference frame and take the base of the rod to be fixed, with the housing accelerating, however will modelling the dynamic system as static be a reasonable approximation?

Also, given the housing + components is relatively heavy (1-2kg), surely the axial load cannot be ignored? Once the rod starts to deflect, it'll become a pretty significant eccentric load.

 

[Mech_Engineer]

Looking at it in the static case is a good start in my opinion, and you can increase complexity with a dynamic analysis if your findings make it seem necessary.

Making assumptions to simplify your analysis is engineering 101, but only you can decide "what's necessary." Step 1 is to put bounds on how much your beam will deflect so you can decide what else might contribute to the design problem.

 

[Travis_King]

You are way overthinking this. You've got a mass (1-2 kg) at the end of a rod. They are connected, so if the robot accelerates, so does the mass (they are connected after all).

You are correct in that the inertia of the mass will mean it wants to resist that acceleration. That's why F=m*a. Hence the deflection. Unless your rod is so undersized that the thing actually bends, I don't think you'll have to care as much about the axial loading. The bending (and the uplift at the connecting rod mounting point) will be your most significant loads.

 

[Mech_Engineer]

You also might calculate the first mode of virbration for the beam/mass, because that number will give you some insight into how stiff it is and how it will hold up to acceleration applied over a certain time constant.

 

[rafehi]

Have always struggled reducing a real world mechanical problem down to a simple model. Working on it...

Just to clarify with a FBD of the external forces:
http://img259.imageshack.us/img259/1201/84664287.jpg

However, assuming that the column isn't going to buckle under the load (not going to happen), the bending caused by the axial force is relatively minor compared to that caused by the acceleration, due to the much smaller moment arm, therefore Fy can be disregarded.

 

[Mech_Engineer]

However, assuming that the column isn't going to buckle under the load (not going to happen),

That's an easy calculation to do too.

the bending caused by the axial force is relatively minor compared to that caused by the acceleration, due to the much smaller moment arm, therefore Fy can be disregarded.

What "bending caused by the axial force" are you referring to exactly? A purely axial force does not cause a moment, is the load not centered on the beam? The acceleration will impart a small moment at the end of the beam (unless you assume the C.G. of the mass is attached to the beam), but you can take that into account if you like with a moment and force applied at the end of the beam... but as you said it will probably be much smaller than the bending moment at the base of the beam.

 

[Travis_King]

He's talking about additional bending load on the rod when the mass is deflected. The mass no longer acts axially, and as a result the mass alone will create additional moment on the rod. As we've said, though, unless we are talking about large deflections here (in which case your rod is too flexible), this should be treated as minimal (unless you want a fully detailed, dynamic result)

 

[Mech_Engineer]

Gotcha, yes I would assume that the deflection on the rod is small. Remember that the force would be subject to sin(theta), so if the beam deflects 1 degree only about 2% of the weight is contributing to a bending moment. If your beam deflection is significant enough to contribute to additional bending moment at its base from a vertical force, I would recommend stiffening the member.

Take a look at the first vibrational mode of the beam/mass system and get back to us.