Mechanics question ;-) Forces on wheels…

[Hodgkinson]

Just a little something for those of you with a mechanics-related inclination ;-)



Take, for example, a Catherine wheel-type firework, as in diagram 1. The 4 motors positioned at the tips of the wheel in the diagram each produce thrust in the direction of the blue arrows, thus causing the wheel to rotate in the opposite direction. Fair enough so far.

Now, connect the Catherine wheel to a drive wheel on some sort of buggy of the same diameter, as in diagram 2. As the Catherine wheel spins, torque (T) is transferred along the shaft to the wheel. Since the wheel converts rotational torque into linear motion along the ground (The brown lines), force R, then this seems to infer that the individual contribution of the thrust from each of the 4 motors to the torque output, and thus the motion of the buggy is always constant and even…

Now here’s the other way of looking at it. Since the Catherine wheel is of the same physical size as the driven wheel and the drive shaft is simply serving as a connecting medium, then the driven wheel can therefore be replaced by the Catherine wheel (neglecting thrust interactions to the buggy, etc), as shown in diagram 3. In diagram 3A, the motor thrust at the top of the wheel acts on the wheel in the form of a second class lever, with the fulcrum point (F) at the ground and the force output forcing the axle of the buggy forwards (Force R).
In diagrams 3B and 3C, a similar scenario is present compared to 3A, except the theoretical levers are bent at 90 degrees.
However, in diagram 3D, the motor thrust is acting at the fulcrum point, and as a result, presumably this motor produces no forwards force on the buggy.

Hence, why does there appear to be a mechanical difference between these two viewpoints, and if there really is a difference, could the motors passing nearest to the ground be shut off in order to improve fuel efficiency?

(Why am I asking this? For those of you who have seen Dad's Army, can you remember the rolling, rocket-propelled mine type contraption? What’s their proper name?)

Hodgkinson.

[JaXanim]

You need to consider the moment of each force at the fulcrum in each scenario. The moment is the product of the force applied and the distance from the fulcrum.
In the catherine wheel, each jet produces the same moment, so if the radius of the wheel is 'r' and the thrust/force is 'f', then the total moment is 4fr.
In the other case, the moment of each jet varies according to its position relative to the fulcrum. The range is between 2fr and zero per jet. So in the rolling wheel case the jets have twice the turning moment when located at the top and zero moment at the bottom. Assuming no other losses, the net turning force remains the same in each scenario.

JaX

[ZeBeeDee]

Hodgkinson wrote:
(Why am I asking this? For those of you who have seen Dad's Army, can you remember the rolling, rocket-propelled mine type contraption? What’s their proper name?)

Hodgkinson.

You mean ... this  :-D

(and the real name is a panjandrum)

[Hodgkinson]

Thanks for the link! (I think that explains why none of my searches turned up anything useful...)

It seems unusual that simply by omitting the virtual connecting rod and placing the wheel directly on the ground that, from each of the rockets producing a constant moment that suddenly we find that the rockets passing nearest the ground have little effect on the motion of the wheel, whereas those positioned furthest away have the greatest effect.

[JaXanim]

Hodgkinson wrote:
It seems unusual that simply by omitting the virtual connecting rod and placing the wheel directly on the ground that, from each of the rockets producing a constant moment that suddenly we find that the rockets passing nearest the ground have little effect on the motion of the wheel, whereas those positioned furthest away have the greatest effect.

The lower jet still has moment about the centre (as they all do), but that would only be apparent if the contact between the wheel and the ground were less than 100% efficient. In other words, if the wheel could slip due to low frictional contact, then it would tend to skid round and tend towards the free catherine Wheel motion. You can visualise the work done by each jet in driving the wheel by plotting the locus of the point as the wheel rolls. I believe you'll see a sine curve - or is it series of semi-circles(?).
(work = force x distance travelled)

JaX

[Oliver]

Don't forget each jet also applies a force to the body, not just resolving to a torque. Just remember that all forces (including reaction forces) and torques must be resolved in order to determine the net. It doesn't matter how you choose to resolve them, but some methods simplifiy the means to a solution.

Turning off the rocket which is next to the ground, does not resolve to the same net result. Remember that a balanced torque couple has no net force on the free body.

If you are interested in this sort of thing, then a first year text book on mechanical engineering dynamics will be a good read for you. Usually, the principles will be covered in the first couple of chapters. Personally, I found this sort of material really fun to study, and intuitively satisfying.

[Zac67]

JaXanim wrote:
I believe you'll see a sine curve - or is it series of semi-circles(?).

Definitely semi-circles, diameter equivalent to the wheel's circumference. For a sine you'd have to plot the y(t) coordinate only; in combination with the center-relative x(t) (=real position in space) you'll get semi-circles (double speed at the top, no speed at the bottom, touching the ground once every revolution).