Calculating ideal wheel radius for submersible vehicle

[Upsilon]

I have an engineering project to do this semester. I'm not going to get into the specifics, but I (and my group) are going to be building a submersible servo-driven vehicle (it is basically driving underwater). The vehicle will be made of 4" PVC piping (thin-walled sewer variant). It must be able to carry an orange as cargo (don't ask why). Here is a poorly drawn MS Paint top-down view of the proposed design. The T-shaped section is the part made of PVC. The bottom of the T is a screw-cap to allow access to the internal computer and allow the orange to be placed inside, as well as extra ballast. The two "wings" of the T (that have the small flat rectangles on the ends) are permanent flat caps with waterproof servos built in. The small flat rectangles are supposed to be the wheels. The large rectangle on the front is a sort of hydrofoil (it really is just going to be a bent piece of sheet metal duct-taped to the front). The direction of motion is denoted by the arrow.

The task at hand is to determine what wheel radius will provide the highest top speed for the vehicle. Two servos are in operation, providing 80 oz-in of torque each, with a transit time advertised as 0.23 sec/60* (or 0.72 revolutions every second). I decided to simulate the vehicle's motion in Excel to accomplish this, so I could easily manipulate all the different variables involved, and to be able to see exactly how long it will take the vehicle to go a certain distance (acceleration factored in).

The key to this is solving for the acceleration as a function of velocity, since acceleration is affected by drag which is a function of velocity. My mathematical procedure is here, and the variable reference is here (I understand that some of these terms are incorrectly defined, but I did this for the sake of ease of explanation to some of my group members who have less experience in physics. They are technically correct for this application). Apologies in advance, all units are imperial (feet, pounds, slugs) - it was more convenient to do it this way since the servo specs were in imperial units. I started with the amount of ballast I would need to load onto the vehicle, both to counteract buoyancy and to maintain traction with the ground. From there you just calculate the mass of the vehicle and then you can solve for acceleration. If someone could double-check to make sure I did this correctly, I would greatly appreciate it.

In addition, I would like to have a ballpark estimate for the drag coefficient. In my Excel spreadsheet, I'm assuming 0.8 which sounds reasonable to me, but I really do not know for sure. The sheet metal hydrofoil on the front is bent at around 30 degrees.

Other than that, in my Excel spreadsheet, I used 0.6 for the coefficient of static friction (rubber wheels on a wet concrete floor), a 0.161 ft^3 vehicle volume (for buoyancy calculation), and a reference area of 0.391 ft^2 (for drag calculation). With these parameters, I am getting that the ideal wheel radius is approximately 6 inches, with a top speed of ~2.35 ft/sec.

 

[berkeman]

Trying to use wheel friction for underwater propulsion seems non-optimal to me. Why are you trying to do this for an *engineering* project? What other means of underwater propulsion would be so much more efficient?

 

[Simon Bridge]

ATVs are sometimes like that ... so a vehicle may need to be mostly on land but need to traverse arbitrarily deep bits of water. Anyway: projects that are assigned can be pretty arbitrary. I suppose the next question is: how did you end up choosing wheeled propulsion along the bottom for this project?

I have an engineering project to do this semester.

The task at hand is to determine what wheel radius will provide the highest top speed for the vehicle.

The key to this is solving for the acceleration as a function of velocity, since acceleration is affected by drag which is a function of velocity. My mathematical procedure is here, and the variable reference is here

... the equations don't mean anything without comments.
Please describe your reasoning.

(I understand that some of these terms are incorrectly defined, but I did this for the sake of ease of explanation to some of my group members who have less experience in physics. They are technically correct for this application).

You are dumbing things down for the sake of ignorant people ... a common mistake. Either educate them, or work out another approach. i.e. do you have to do a calculation?

I started with the amount of ballast I would need to load onto the vehicle, both to counteract buoyancy and to maintain traction with the ground. From there you just calculate the mass of the vehicle and then you can solve for acceleration. If someone could double-check to make sure I did this correctly, I would greatly appreciate it.

... you modeled the vehicle using a FBD? You have mass, bouyancy, and friction ... (do you realize friction is what drives the vehicle forward), and drag. You don't have a good model for friction (the coefficient depends on the surfaces) or drag (which depends on speed, sure, however the coefficient depends on the geometry of your vehicle, including the wheel size and maybe the tread). The buoyancy will also have a contribution from the wheel.

In addition, I would like to have a ballpark estimate for the drag coefficient. In my Excel spreadsheet, I'm assuming 0.8 which sounds reasonable to me, but I really do not know for sure. The sheet metal hydrofoil on the front is bent at around 30 degrees.

... so you also get drag from the front hydrofoil, as well as some extra ... um... what is the opposite of lift(?) that also depends on speed. Technically you could add a hydroscrew (propeller) to push the wheels harder against the bottom ;)

I think there are too many variables: define "correct". Making a model for the behaviour and representing that model as maths which you study using a computer is "correct" procedure.

Other than that, in my Excel spreadsheet, I used 0.6 for the coefficient of static friction (rubber wheels on a wet concrete floor), a 0.161 ft^3 vehicle volume (for buoyancy calculation), and a reference area of 0.391 ft^2 (for drag calculation). With these parameters, I am getting that the ideal wheel radius is approximately 6 inches, with a top speed of ~2.35 ft/sec.

You know the surface will be concrete? Do the tyres have to be rubber? Do you have to have fixed ballast?
Are you expected to build the ideal wheel or just pick a size from stock?

 

[Upsilon]

Trying to use wheel friction for underwater propulsion seems non-optimal to me. Why are you trying to do this for an *engineering* project? What other means of underwater propulsion would be so much more efficient?

I agree that it is probably not the best, but we are not allowed to use compressed gases, so the only other option would be a propeller. I chose powered wheels over the propeller just for ease of operation and design (this is an entry-level freshman year project might I add, not a senior design project).

ATVs are sometimes like that ... so a vehicle may need to be mostly on land but need to traverse arbitrarily deep bits of water. Anyway: projects that are assigned can be pretty arbitrary. I suppose the next question is: how did you end up choosing wheeled propulsion along the bottom for this project?

That isn't the case here - the vehicle will be strictly underwater. I described why I chose this design above.

... the equations don't mean anything without comments.
Please describe your reasoning.

I was hoping that the variable reference would be enough to explain, but I guess not. Very well, I'll go through it line-by-line:
Line 1: Maximum force of static friction
Line 2: Drag induced by the water as a function of velocity
Line 3: Maximum total propulsion force, given by twice the torque provided by a single servo (since there are two of them) divided by the wheel radius
Line 4: Buoyant force as determined by the volume of the vehicle, this will need to be counteracted with ballast
Line 5: Assumption that in order to maintain wheel traction during the acceleration phase, the maximum static frictional force must be greater than or equal to the force of propulsion provided by the servos. I later just resolve this to an equality, assuming that the most efficient setup is to have static frictional force equal to propulsion force (loading the minimum amount of extra weight required to maintain traction).
Line 6: Total weight of the vehicle, defined by the force needed to counteract buoyancy (equal to the buoyant force disregarding direction) plus the force needed to maintain static friction (or the normal force)
Line 7: Calculation of the mass of the vehicle given its total weight on dry land divided by the gravitational constant
Line 8: Calculation of the net forward force acting upon the vehicle, given by propulsion force minus force of drag
Line 9: Calculation of the forward acceleration of the object, given by the net forward force divided by the vehicle's mass

... you modeled the vehicle using a FBD? You have mass, bouyancy, and friction ... (do you realize friction is what drives the vehicle forward), and drag. You don't have a good model for friction (the coefficient depends on the surfaces) or drag (which depends on speed, sure, however the coefficient depends on the geometry of your vehicle, including the wheel size and maybe the tread). The buoyancy will also have a contribution from the wheel.

I did not explicitly create a FBD, but I am aware of all these things. In my Excel spreadsheet, I left the coefficient of friction and drag coefficient as modifiable variables so that I could simulate the vehicle's motion under different conditions. The small nuances you mention as a result of the wheel will just make the computation that much more complex, so I figure the best approach would be to leave them out of the computation and assume a margin of error (i.e. the computation says a 6" radius wheel is ideal, so I would go for a 5" radius wheel).

... so you also get drag from the front hydrofoil, as well as some extra ... um... what is the opposite of lift(?) that also depends on speed. Technically you could add a hydroscrew (propeller) to push the wheels harder against the bottom ;)

That is a good idea...while it won't really improve the acceleration at all, it will at least reduce the amount of ballast we need to load on.

I think there are too many variables: define "correct". Making a model for the behaviour and representing that model as maths which you study using a computer is "correct" procedure.

That is what I have done in Excel - it is essentially a simulation of the vehicle's motion as a result of all these different variables. As I mentioned above, the spreadsheet takes in coefficient of friction, drag coefficient, vehicle volume, and reference area. Changing these values will adjust the results of the simulation accordingly.

You know the surface will be concrete? Do the tyres have to be rubber? Do you have to have fixed ballast?
Are you expected to build the ideal wheel or just pick a size from stock?

The surface will definitely be concrete. The tires don't need to be made of any specific material, but I figured that rubber would be the best for this purpose because it tends to form high coefficients of friction with other materials. It doesn't matter how we come by the wheel, but I think we'll probably just make it because I can't seem to find any stock servo wheels in the sizes we're talking about here. I'm thinking of using 10" frisbees and wrapping a rubber band around the outside edges for traction. The rubber bands would be glued in place so they don't slip off.

 

[OldYat47]

Very generally, the smallest radius practical would be beneficial in three ways. First, it would present a relatively smaller contact patch so hydroplaning would be less of an issue. Second, the normal force would be higher in the area of contact. Third, a smaller wheel (radius and width) produces less drag. Actual drag and friction would dictate the optimum gearing for the torque available.