Maximum weight carried by a specific torque

[Hugh Carstensen]

Hello,

My name is Hugh Carstensen. I am a CSE undergrad at the Ohio State University.

I recently secured a position designing and assembling an automated camera-rig for digitization of archival works in the Knowlton School of Architecture.

The rig will be powered by a number of small stepper motors set on X and Y carriers, respectively. In trying to decide which steppers are adequate for the job, it came about that a calculation would have to be made in order to find the maximum weight that any given stepper could move effectively (based upon the output torque of the given steppers.

It has been some time since my last physics class (I'm a CSE major), and I would like if someone would be so kind as to review my calculations for accuracy.

Here is the basic idea:
In order to move the load of the cart, I need to compare the force exerted by the motor (F_m) with the friction force (F_f).

Concerned formulas (Some of these are simple, because I believe certain more complex elements are negligible on smaller scales):
F_m ≥ F_f
F_f = μ × F_n
F_n = m × g × ½*

*Each carrier will have 2 wheels

F_m = τ / r

F_m - Force exerted by the motor
F_f - Force exerted by friction
F_n - Normal force
μ - Coefficient of friction (rolling resistance in this case)
g - Acceleration of gravity
m - The mass that I am solving for
τ - Torqe of the stepper
r- radius of the wheel attached to the stepper

My calculations (Using 20.0 N×cm for torque in this example):
F_f = μ × F_n
F_f = μ × (m × g × ½)
F_m ≥ F_f = μ × (m × g × ½)
τ / r ≥ μ × (m × g × ½)

20.0 N×cm / 1.27 cm ≥ .006* × m × 9.80m/s² × ½

*.006 is an approximation of the rolling resistance of rubber on smooth aluminum

15.7 N ≥ 2.94 m/s² × m
5.34 Kg ≥ m
or conversely
m ≤ 5.34 Kg

Which, if I am correct, would mean that two steppers with wheels of radius 1.27cm can move a load of 5.34Kg or less.

Now for the fun part. What did I do wrong? (If there is anything, of course).
Furthermore, is this answer accurate enough?
Is there a more precise method with which I should approach this problem?

Thanks,

Hugh

 

[Nidum]

Best to start over I think .

If as it seems these carts are used positioning equipment with high accuracy and with frequent moves you really need to be evaluating the effects of inertial loading and designing your control system to incorporate things like soft start and stop velocity profiles .

Post a picture or good drawing of this rig and tell us as much as you can about required accuracy of positioning , speeds of travel , frequency of moves et al .

 

[Hugh Carstensen]

Best to start over I think .

If as it seems these carts are used positioning equipment with high accuracy and with frequent moves you really need to be evaluating the effects of inertial loading and designing your control system to incorporate things like soft start and stop velocity profiles .

Post a picture or good drawing of this rig and tell us as much as you can about required accuracy of positioning , speeds of travel , frequency of moves et al .

I can't make a scan of the drawing until my supervisor has finished creating the rectified version. I will say however that the cart need not be incredibly precise (the previous prototype was driven using a handle. (The paper describing the previous version can be found here: https://osu.app.box.com/s/zw4p2qx2jlg7a2sr5heg29i47xmdk2dd). The idea of this project is to create an automated version of the same design, with materials that don't bow (as easily) under the weight of the camera. The carts will ideally move very slowly (for more accurate image stitching). The program operating the motors will also use acceleration/de-acceleration sequences in order to prevent any skipping, though it's maximum velocity is likely never to exceed 1.2cm/s. The moves will not be consecutive; essentially long passes with the small carrier, a pause for the image to stitch, then a short move by the larger carrier, a pause to begin image collection, and another long move from the smaller carrier. Accuracy need only be within ~1/4 of an inch.

 

[tygerdawg]

Need a sketch, else you're wasting everyone's time. Draw freehand with pencil & paper, scan it, post it. Is the X-Y motion in a vertical plane, and that is why you are considering gravity effects? Or is the motion in a horizontal plane? If horizontal, no gravity effects. Sloped, then proportional gravity effects. A sketch would clarify.

I ***glanced*** at the math, you are generally on the right track. Websearch, find, download a document called Smart Motion Cheat Sheet in PDF, it has a good summary of necessary equations.

Motor should be sized for peak torque, which is torque needed for acceleration (speed up/down), overcoming translational & rotational friction, moving translational & rotational load (inertias) at steady state, several other things. You can be lazy, do fat & lazy analysis and buy oversized motors to make sure you have enough.

Just an aside note: do I read this correctly? Your mechanics will be wheels in some sort of track or something? Not advisable for any kind of robustness or accuracy. The rig should be set in some sort of linear rail bearings if your time & budget allows. See something like www.pbclinear.com for examples.

 

[Hugh Carstensen]

Need a sketch, else you're wasting everyone's time. Draw freehand with pencil & paper, scan it, post it. Is the X-Y motion in a vertical plane, and that is why you are considering gravity effects? Or is the motion in a horizontal plane? If horizontal, no gravity effects. Sloped, then proportional gravity effects. A sketch would clarify.

I ***glanced*** at the math, you are generally on the right track. Websearch, find, download a document called Smart Motion Cheat Sheet in PDF, it has a good summary of necessary equations.

Motor should be sized for peak torque, which is torque needed for acceleration (speed up/down), overcoming translational & rotational friction, moving translational & rotational load (inertias) at steady state, several other things. You can be lazy, do fat & lazy analysis and buy oversized motors to make sure you have enough.

Just an aside note: do I read this correctly? Your mechanics will be wheels in some sort of track or something? Not advisable for any kind of robustness or accuracy. The rig should be set in some sort of linear rail bearings if your time & budget allows. See something like www.pbclinear.com for examples.

First let me explain the diagram above:
Green: Rails (upon which each carrier rides).
Purple: Non-rail pieces of frame.
Blue: Linear bearings & motor blocks.

The device sits horizontally. I was using gravitational acceleration in the expression for normal force. The issue is that I would calculate for torque if there were a reliable way to figure the mass of each carrier, which will fluctuate depending on the camera in use (which sits in the middle carrier), and the lighting in use. As a result, I have chosen to find a maximum mass that can be carried by a motor and have used 20 N×cm for the purpose of demonstration (in the math above).

Yes, the mechanics in the system are driven via steppers on rails. However, they are paired with linear bearings (created using cast steel bearings and C-channel). In order for this to make sense, allow me to describe the restrictions placed on the design of this machine:

1. The device must be expandable in order to work with images of various (large) sizes.

This is because the device is intended to be portable and reconfigurable; it will often be used on-site.
As such, the largest rails (as well as those carried by the larger carrier) are broken into 2-foot sections which are held together by a system of coupling braces (such that the top of the rail is smooth).

2. The device must be made with low cost materials.

This device need be budget friendly such that an archival department anywhere can afford to create their own.

Precision, again, is not paramount, the way that the device will function leaves margins on the edge of each photograph composing the whole scan, the program that stitches these images works well with margins, and thus can cope easily with a fairly large degree of imprecision.

 

[deuel18]

Have u looked at xyz core 3d printers? Ur set up is very similar to its mechanism. Depending on the mounting of the camera on the moving profile, u do need to know its inertia.

 

[Hugh Carstensen]

Have u looked at xyz core 3d printers? Ur set up is very similar to its mechanism. Depending on the mounting of the camera on the moving profile, u do need to know its inertia.View attachment 210077

Yes, that is very similar, apart from being vertical.

I'll have to work out the inertia, but as a bit of helper, I have been coding a fairly robust PID controller for each of the carriers, with a large enough derivative gain, the carrier should speed up and slow down well.

Also, the motors are now servos instead of steppers due to cost restrictions.