Base sizing of a robotic arm for maximum stability

[Aaron Mac]

Homework Statement

I wanted to size the circular base at the bottom for maximum stability so the the whole system doesn't topple over and fall when robotic arm is under operation.

i tried many methods but i am not convinced with the results and equations i am trying.

Relevant Equations

Bending moment = (Weight of steel tube + weight of robotic arm) x Height of steel tube

Or maybe Stability factor = 2

 

[Baluncore]

Assuming the base will be anchored with eight peripheral bolts.
I think you need four triangular gusset plates between the column and the circular base.

 

[Aaron Mac]

Hello friend, it is already in the original 3d model. I'm just having a hard time sizing the base.

 

[Baluncore]

I'm just having a hard time sizing the base.

How is the base attached to the floor or foundation, bolts ?

 

[Aaron Mac]

Yes right i have also calculated the clamping force per bolts.

 

[Aaron Mac]

Assuming the base will be anchored with eight peripheral bolts.
I think you need four triangular gusset plates between the column and the circular base.

I did consider triangular gusset plates initially but due to the functionality of the robotic arm it won't be possible. So it was decided to design a base which will be clamped onto the floor with bolts

 

[Baluncore]

Then the base thickness will be decided by the steel tube wall thickness, and the quality of the weld, or pipe thread joint.

 

[Aaron Mac]

i decided to use an 8mm base thickness as mentioned in the paper and we can ignore the quality of the weld if it can make the base sizing much simpler. Can you guide me through the calculations?

 

[Baluncore]

Have you decided the size, and the number of bolts, appropriate for an 8 mm plate ?
The bolt heads must not pull through the plate.
The threads should not strip before the plate bends.

Imagine the 1-metre-long tube is a lever arm. What side force will that carry?
Compute the moment where the arm meets the base. Make the base diameter sufficient to not pull the bolts out of the floor when subjected to that force.

 

[Aaron Mac]

By side force you mean weight of the robotic arm and payload it will be carrying? I did account that when calculating the bending moment at the bottom. And how would you size the base diameter and mass?
I am a little bit lost honestly!!
Thanks in advance

 

[Baluncore]

I did account that when calculating the bending moment at the bottom.

That bending moment, at the bottom of the tube, will be transferred to the base diameter by the joint. The base must not overturn when that moment is applied to the base.
You will need to know the strength of the bolts that match the 8 mm thick base plate.

 

[Aaron Mac]

That bending moment, at the bottom of the tube, will be transferred to the base diameter by the joint. The base must not overturn when that moment is applied to the base.
You will need to know the strength of the bolts that match the 8 mm thick base plate.

you mean the clamping force of the bolts to see the consequence on the base plate?
Do you have any particular equation for that?

 

[Baluncore]

Do you have any particular equation for that?

You need to understand the moment, rather than have an equation.
You know the moment at the bottom of the 1-metre-long column, will be transferred to the base diameter, because they are joined together, and so form a lever arm.

If you knew the ratio of the base diameter, to the column length, you could work out the base diameter.

You know the force that generates the moment on the known length column, and you know the tensile strength of the bolts. The ratio of those, is the ratio you need, to work out the base diameter.

 

[Aaron Mac]

"If you knew the ratio of the base diameter, to the column length, you could work out the base diameter."
The length of the column is already known as a consequence. If we assume a ratio we are essentially assuming the diameter already

 

[Baluncore]

If we assume a ratio we are essentially assuming the diameter already

That would be true if you assumed the ratio.
But no assumption is necessary, you can find it in another way.
The ratio is the same as the ratio of "moment causing force", to the "strength of the bolts".

 

[Aaron Mac]

Ratio = Moment causing force / (Strength of bolts * Cross-sectional area of the column)

Is this the appropriate ratio equation?

 

[Lnewqban]

By side force you mean weight of the robotic arm and payload it will be carrying? I did account that when calculating the bending moment at the bottom. And how would you size the base diameter and mass?
I am a little bit lost honestly!!
Thanks in advance

The bending moment at the I-beam weld to the base should not include the height of the column (except for accidental and any dynamic horizontal loads), only the weight and load of the arm and its horizontal distance to the center of the base.

It seems to be a little moment based on the represented 25 Newtons.
What that 8 mm steel base is going to be anchored to? An existing concrete slab? A footing to be built? A steel structure?

 

[Aaron Mac]

it will be fixed onto concrete. It is actually a c channel and you are right about the bending moment calculation; i did that for safety factor since in real practice it may not be welded onto the neutral axis and dynamic loads etc.
can you help me size the base, i honestly spent days trying to do it but in vain no good results in the end.

Thanks in advance

 

[Lnewqban]

it will be fixed onto concrete. It is actually a c channel and you are right about the bending moment calculation; i did that for safety factor since in real practice it may not be welded onto the neutral axis and dynamic loads etc.
can you help me size the base, i honestly spent days trying to do it but in vain no good results in the end.

Thanks in advance

Follow the advice provided by @Baluncore above.

You will have a calculated pair of forces acting on the base of the C-channel.
Use the less advantageous orientation of the arm respect to the smaller dimension of the cross-section of the channel, as that will impose greater bending efforts on the weld and anchors.

Consider that the plate will deform under bending stress, more as more separated are the anchors from the neutral axis.
That deformation will be small there, but the column and the arm will magnify it and the tip of the arm and the load may bounce up and down noticeably.

Select the size of your anchors also according to what you have available, and to how deep the embedment into concrete can be, without reaching the reinforcing steel within it, all around your circle of perforations.

 

[Aaron Mac]

I appreciate the advice but i wanted to provide justifications for the size and mass of the base

 

[erobz]

I appreciate the advice but i wanted to provide justifications for the size and mass of the base

The justification for the size of the base is at least partially in the distance of the bolts from the point about which the base will tip. The bolts that are further away tend to have load proportional to that distance via applying elastic theory assuming the plate is practically ridged. The plate itself obvious can't be made of aluminum foil, otherwise it is not going to be practically ridged under the loading, but it also may not need to be the stiffness(strength) of a pile anchoring the Empire State Building.

Assume elastic deformation of the bolts in this figure to balance the applied moment $M$ from the load, about the tipping point $A$. What do you find?

 

[Aaron Mac]

Please check this below and let me know if it is okayTo determine the load on the bolts, we can use the principle of statics, which states that the sum of all forces and moments acting on a rigid body must be zero. Assuming that the plate is rigid, we can balance the applied moment from the load about the tipping point A using the elastic deformation of the bolts.

Let's assume that the distance from the bolts to the tipping point A is L. We can also assume that the plate is supported by N bolts, each with a diameter of D. The applied moment from the load is M, and the Young's modulus of the bolts is E.

The elastic deformation of each bolt can be calculated using Hooke's law, which states that the stress is proportional to the strain. For a bolt under tension, the stress is given by:

Stress = Force / Area

where Force is the tension force on the bolt and Area is the cross-sectional area of the bolt.

The strain is given by:

Strain = Deformation / Length

where Deformation is the change in length of the bolt and Length is the original length of the bolt.

The tension force on each bolt can be calculated as:

Force = Load / N

where Load is the total load acting on the plate.

The cross-sectional area of each bolt is given by:

Area = (pi/4) x D^2

The deformation of each bolt can be calculated using the formula for elongation due to tension:

Deformation = (Force x L) / (E x Area)

The moment generated by the deformation of each bolt is given by:

Moment = Force x L

The total moment generated by all the bolts is:

Total moment = N x Moment

Substituting the expressions for Force, Area, Deformation, and Moment, we get:

Total moment = N x (Load / N) x L x ((Load / N) x L / (E x (pi/4) x D^2))

Simplifying, we get:

Total moment = (Load x L^2) / (E x (pi/4) x D^2)

To prevent the plate from tipping, the total moment generated by the load must be balanced by an equal and opposite moment generated by the bolts. This means that the maximum moment that the bolts can sustain is:

Maximum moment = Total moment / Safety factor

where the safety factor accounts for uncertainties in the load and the strength of the bolts.

The maximum stress on the bolts can be calculated using the formula for bending stress:

Bending stress = (Maximum moment x Distance from neutral axis) / (Section modulus x Safety factor)

Assuming a circular cross-section for the bolts, the section modulus can be calculated as:

Section modulus = (pi/32) x D^4

Solving for the bolt diameter, we get:

D = (32 x Maximum moment x Distance from neutral axis) / (pi x Bending stress x Safety factor x E)^0.5

Substituting the expressions for Maximum moment, Distance from neutral axis, Section modulus, and Bending stress, we get:

D = (32 x (Load x L^2 / E) x L / (pi x (Load x L^2 / E) x Safety factor x (pi/32) x D^4)) ^ 0.25

Simplifying, we get:

D = (256 x L^3 x E / (pi^3 x Load x Safety factor)) ^ 0.2

Assuming a Young's modulus of 200 GPa for the bolts, a load of 1000 N, a distance from the tipping point of 0.2 m, and a safety factor of 2, we get:

D = (256 x (0.2 m)^3

 

[erobz]

Please check this below and let me know if it is okayTo determine the load on the bolts, we can use the principle of statics, which states that the sum of all forces and moments acting on a rigid body must be zero. Assuming that the plate is rigid, we can balance the applied moment from the load about the tipping point A using the elastic deformation of the bolts.

Let's assume that the distance from the bolts to the tipping point A is L. We can also assume that the plate is supported by N bolts, each with a diameter of D. The applied moment from the load is M, and the Young's modulus of the bolts is E.

The elastic deformation of each bolt can be calculated using Hooke's law, which states that the stress is proportional to the strain. For a bolt under tension, the stress is given by:

Stress = Force / Area

where Force is the tension force on the bolt and Area is the cross-sectional area of the bolt.

The strain is given by:

Strain = Deformation / Length

where Deformation is the change in length of the bolt and Length is the original length of the bolt.

The tension force on each bolt can be calculated as:

Force = Load / N

where Load is the total load acting on the plate.

The cross-sectional area of each bolt is given by:

Area = (pi/4) x D^2

The deformation of each bolt can be calculated using the formula for elongation due to tension:

Deformation = (Force x L) / (E x Area)

The moment generated by the deformation of each bolt is given by:

Moment = Force x L

The total moment generated by all the bolts is:

Total moment = N x Moment

Substituting the expressions for Force, Area, Deformation, and Moment, we get:

Total moment = N x (Load / N) x L x ((Load / N) x L / (E x (pi/4) x D^2))

Simplifying, we get:

Total moment = (Load x L^2) / (E x (pi/4) x D^2)

To prevent the plate from tipping, the total moment generated by the load must be balanced by an equal and opposite moment generated by the bolts. This means that the maximum moment that the bolts can sustain is:

Maximum moment = Total moment / Safety factor

where the safety factor accounts for uncertainties in the load and the strength of the bolts.

The maximum stress on the bolts can be calculated using the formula for bending stress:

Bending stress = (Maximum moment x Distance from neutral axis) / (Section modulus x Safety factor)

Assuming a circular cross-section for the bolts, the section modulus can be calculated as:

Section modulus = (pi/32) x D^4

Solving for the bolt diameter, we get:

D = (32 x Maximum moment x Distance from neutral axis) / (pi x Bending stress x Safety factor x E)^0.5

Substituting the expressions for Maximum moment, Distance from neutral axis, Section modulus, and Bending stress, we get:

D = (32 x (Load x L^2 / E) x L / (pi x (Load x L^2 / E) x Safety factor x (pi/32) x D^4)) ^ 0.25

Simplifying, we get:

D = (256 x L^3 x E / (pi^3 x Load x Safety factor)) ^ 0.2

Assuming a Young's modulus of 200 GPa for the bolts, a load of 1000 N, a distance from the tipping point of 0.2 m, and a safety factor of 2, we get:

D = (256 x (0.2 m)^3

Slow down a bit. Your bolts are not all equidistant from an axis passing through the tipping point. Show me the equation for the sum of the moments in just the simplified diagram I presented. Just two bolts for now to wrap your head around what I'm talking about.

 

[Aaron Mac]

Sum of moments = F x (L1 + L2)

 

[erobz]

Sum of moments = F x (L1 + L2)

No (that's not quite correct), the forces developed in each bolt are different.

 

[erobz]

Hint: You aren't going to be able to solve anything just yet ( too many variables, not enough equations at the moment), we are just writing down the base equation to work with.

Also...I understand you're in a hurry, but could you take a moment to learn Latex. These equations are fairly simple to interpret, but you've posted on PF several times now. It makes it much easier for people that wish to help, examine your math.

 

[Aaron Mac]

I am honestly working on my phone, and i sincerely apologise. I will make sure next time i follow your advice :)
I think maybe i'm getting what you are trying to do. It is to balance sum of moments of all bolts with the bending moment at the bottom due to the robotic arm?
if yes, then what next?

 

[erobz]

I am honestly working on my phone, and i sincerely apologise. I will make sure next time i follow your advice :)
I think maybe i'm getting what you are trying to do. It is to balance sum of moments of all bolts with the bending moment at the bottom due to the robotic arm?
if yes, then what next?

Then you let the bolts deflect some small amount $\delta_i$ assuming the plate stays "straight-rigid member" as it "tips". Relate the deflection of bolt $l_1$ to the deflection of the bolt at $l_2$ using that geometry and the force $F_1$ to the force $F_2$ via the elastic equations for each bolt.

 

[Aaron Mac]

Then you let the bolts deflect some small amount $\delta_i$ assuming the plate stays "straight-rigid member" as it "tips". Relate the deflection of bolt $l_1$ to the deflection of the bolt at $l_2$ using that geometry and the force $F_1$ to the force $F_2$ via the elastic equations for each bolt.

How will I find the base size and mass from your explanations?

 

[erobz]

How will I find the base size and mass from your explanations?

Are you going to make the base so massive that you don’t need to bolt it to the floor, or are you going to make it strong enough to withstand the loads acting on it and let the bolts hold it to the floor? There is no “right”answer IMO. It’s something you need to “think” about. What I am trying to show you is how to calculate the stress in the bolt from the loading on the base. Which in turn is a loading on the base. Your job would then be to the make sure the base is strong enough to remain effectively rigid under said loading. Others mentioned gussets, and the floor, etc…all these things must be given some careful thought. I’m afraid formula grabbing your way through this is not going to work out so well.