So I've had an idea for a special quarterstaff with a solid knob on each end and I was wondering if and how practical it would be in a real fight. I did some research, and the closest I could find was the African knobkerrie, which looks to be maybe four-ish feet long, but what I really want to know is how a much longer and knobbed on each end staff would balance, and if the resulting weapon would be able to be wielded to a good effect.
Hehe, martial art question in worldbuilding! Let's take historical short staffs and long staffs separately before treating the double knob staff.
The longest staff/staff-like weapons that have been used historically are the pike poles, halberds, the Chinese spear poles and the Japanese naginata. These were mostly not targeted at people but at horses during warfare. These are invariably weighted at one end and are not expected to be balanced at all. The knife or spear point is heavy, bearing these at one of the two ends of a 7-10 feet long pole: the metal blade (for slicing and 'poking' power), the tying rope and the extra tassels etc (to cause maximum internal damage as well as slip out of flesh quickly while ripping it out, brr, boy, am I super glad we don't live in those ages!). This is intentional since the unbalanced end makes it easier to wield - your strength and gravity work together to keep the long staff in constant motion and give it its thrusting power. These poles are usually flexible to prevent breakage. Check out the special martial arts practice routines devised around these weapons to see what I mean. YouTube has several (apols, can't seem to link anything in the post on a mobile device)
There exist double balanced blades on both ends of a long pole but these are not as popular as single ended ones. I suspect the double ended ones would look flashier since it would be easy to do rotational motions and slices with them. But these would need greater strength to thrust and change direction while wielding. (added later:) Also, with a 7.5-foot chinese spear pole, about 3/4ths of the moves hold the pole not at the balance point halfway, but with one hand a bit above one end and the second 1/3rd of the length up, the intention being that there should be 'a good length to poke and slice with'. Any sharp object on the shorter end would be a major irritation and require extra care not to cut yourself with. It can be done, like with a double naginata, just more trouble than it's worth. Since this is a knob, not a knife, doesn't have a cutting problem but might have a manoeuvering problem if the weight is too heavy.
Short staffs usually do not have any spear points and knives at the end. At short range, the points are difficult to wield and attack with while protecting yourself at the same time. So usually the staffs at 4-6 feet length are unweighted at the ends except to take advantage of 'gravity assist', you could have a knob at one end acting as a bludgeon. Staffs at this length are more of blocking and bludgeoning weapons than slicing or thrusting weapons. Also they can be 'fatter' or more inflexible than long poles for the weight of it as a weapon that can break bones and hurt. As an added advantage you can use them for walking sticks, to chase away dogs and hang up washing (haha, but this is exactly what ancient travelers and wandering monks did, staffs are delightfully multipurpose).
So double knobbed staff okay, bludgeons well and rotates well due to the balanced nature. No gravity assist but sheer human strength can compensate for it at 4-5 feet length, as a thrusting weapon as well. No obvious disadvantages. Just keep the knobs not too big or heavy.
Curiously, a double knobbed staff looks a bit like a long bone...
(added after some more thought): The joint where the knob meets the staff pole would be prone to breakage on impact if the knobs are exactly spherical meeting a thin pole. Perhaps the knobs could meld in like the ends of a bone.
In general you need to pick the right we weapon for the right enemy. I don't know if you'd benefit from two knobs against any kind of enemy. A regular quarterstaff can easily break an unarmored skull, for example. Someone with a helmet would at least get a concussion.
I would like to point out the physics as well. Kinetic energy equals mass times velocity squared, so added mass actually does little compared to just swinging the thing a bit faster. The knob would also spread the force of the impact over a larger area of the enemy, which is not desirable when you really want to hurt them.
A knob on each end would also limit the available moves for the one wielding that thing. You couldn't hold one arm closer much closer to one end of the staff. An asymmetrical grab is essential when you want to put good force behind your strike.
All in all, I think there are good reasons why such staffs are not common.
A single ended version of the weapon already exists, it is called a mace. The purpose of placing a weight at one end of the mace is to add extra mass to deliver a crushing blow to the enemy, and add a lever arm to deliver the force to cause critical injuries to an armoured target.
As noted, increases in velocity considerably increase the force of the blow, so we need to essentially strip the weapon down to the minimum needed to function for a fast blow, consistent with the other needs (ability to withstand the shock of impact, ability to bock or parry enemy strikes and so on).
A double ended mace, which is essentially what you are describing, might offer some advantages, but the symmetrical placement of the weights would probably create more problems for the wielder than it is worth.
The weapon would be heavy compared to a conventional staff or one sided mace, so it would be harder to get into action, more difficult to get up to speed for a killing blow, and probably the most critical factor is the moment of inertia would make it difficult to quickly change the trajectory to strike a moving target or rapidly parry a blow. Moment of inertia can be quickly visualized by imagining a figure skater. When she is spinning with her arms extended, she spins relatively slowly, but when she brings her arms in her rotational speed increases even though she has not otherwise changed the amount of torque used to cause the spin. Your double ended mace is the figure slater with her arms extended.
So while there is nothing in principle to stop you from making and using a double ended mace, the weapon doesn't offer enough of an advantage over a conventional staff or mace to really make it worth the trouble.
I've heard of a staff with iron caps on the end for extra weight and striking power(as well as making the main striking points more durable.) Having knobs on the end would serve the same purpose. As far as balance, that could improve the speed and power of each strike when used with the right style of martial art for the weapon.
In a street fight that doesn't include pole arms, firearms or any edged weapon capable of cutting through the staff your weapon would be a very good choice. The main downside are that it may take some extra practice to get familiar with the differences in balance, and shaping a staff like that would be more time consuming and labor intensive than an ordinary staff, or an iron capped staff.