When professionally designed dual axle trailers have only a brake on a single axle, is it usually the front axle? This was the configuration of the OP's trailer but a couple of the aftermarket modified trailers have the single brake on the rear axle.
Including mine. I don't know why the manufacturers do that, but the physics don't seem to bear it out, unless it really doesn't matter in practice, and it's marginally cheaper to put them on the front axle.
Is seems to me that with independent suspension on the trailers axles, when you go from one axle to two axles, each tire in the dual axle trailer would have 1/2 the weight of each tire in the single axle config. Since both traction and the ability of a tire to stop are proportional to weight on the tire, only having brakes on one axle of dual axle trailer would "in theory" only have 1/2 the braking capability of a single axle trailer with brakes. This is probably at least part of the reason a lot of states require brakes on both axles.
That's what I've always assumed. I don't know what the magnitude of the brake torque is as a function of coupler deceleration (surge) loading, but I doubt it makes much difference on dry roads. Probably more of an effect on wet roads. Just a gut feeling with nothing to back it up.
Also, it seems that if you have any tongue weight, there is going to be a higher weight on the front axle than on the rear axle. The higher the tongue weight, the larger the imbalance ie, the higher the tongue weight, the higher the weight on the front axle compared to the rear. So you would want the brakes on the front axle – like the OP’s professionally designed trailer. Also, the braking action likely transfers weight to the front axle - same as for a car - and another reason you would want the brakes on the front axle if you only had the brakes on one axle..
More load on the front axle can normally only happen with independent suspension, and then only if you put the ball at a lower height than the design height of the coupler with the trailer experiencing equal wheel loading. In fact, you can do the opposite by having the hitch ball too high.
Stopping doesn't transfer any weight to the front axle if the trailer isn't allowed to tilt forward while decelerating, and with the rear of the tow vehicle trying to tilt upward (the car
can tilt, however), it's unlikely the trailer attitude will change much if at all even with the tendency of the trailer to rotate forward against the hitch, and therefor there will be essentially no weight transfer to the front axle. Small changes in hitch height, during braking or otherwise, wouldn't have much effect on axle loading, though, as independent (torsion, usually) suspensions are relatively soft for their load rating compared to leaf springs, and with the axles close together but far from the hitch ball, any change in deflection will be small.
In fact, with the trailing arms of a typical torsion suspension, the arms will tend to rotate upward with brake torque, so with brakes on only one axle (either axle), the wheels with brakes will tend to unload a bit as the other axle takes a little more load. Brakes on both axles would cause both to rotate upward a bit, so they'd both maintain essentially the same load. I suspect that's the reason for many (most?) states requiring brakes on all axles.*
Equalized axle suspensions, of course, experience essentially the same static wheel loading no matter what the trailer frame attitude, barring weight transfer due to brake torque at the axle. Below is a pic of my trailer when I first pulled the burned up brakes off when I first got the boat. You can see (coincidentally) how the rear axle would take more load as it rotates forward, and the front takes less, especially if it too has brakes. Front is to the left.
*Edit: You can see this effect with tractor-trailers with pneumatic suspensions with trailing links, where they release their brakes at a light (or just stop) and the tractor or trailer 'stands up' taller. Since all axles have brakes (federal law governs here), the whole rig moves up and down as they all work together, but if only one axle of two or three had brakes, it would unload a bit and the other(s) would take the difference.
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