Let me handwave past a lot of the interrelationship between burn rate, chamber pressure, chamber temperature, thrust, burn cavity, and time. They're all complexly interrelated. If we concentrate solely on the relationship between thrust and cavity, most of the questions can be answered.
We classify thrust profiles broadly as regressive, neutral, or progressive. Regressive means a lot of thrust in the beginning, tapering off. Neutral means roughly the same thrust throughout the burn. Progressive means the thrust increases over time. Different cavity geometries produce different profiles.
About two-thirds of the way down this page
http://www.collectspace.com/ubb/Forum14/HTML/001062.html you can see a cross-section of the Apollo LES motor propellant casting. That's the classic star-shaped cavity, which gives you roughly neutral thrust. This is a great page
https://vc.airvectors.net/tarokt_1.html . It has photos from the rocket park outside the factory where the SRBs are made. (Thiokol, Morton-Thiokol, ATK, Orbital-ATK, and now Northrop-Grumman. I don't care what they call themselves as long as the checks don't bounce.) I added this one because this is a solid propellant casting you can actually touch, if you go there. I've given this tour to people many times. It's a two-hour drive or so from Salt Lake City. Here you can see where someone cut out a chunk of the casting for testing. (This was a casting process test, if memory serves. The "propellant" is actually inert.)
So what about the shuttle? Because the orbiter was so drag-sensitive, the whole STS stack had to throttle back between T+50 and T+70 seconds. How do you throttle back a solid rocket motor and then throttle it back up again? I'm sure you could come up with a convoluted cavity geometry to do that, but then it becomes a manufacturing problem. It's hard enough to cast these giant segments with simple cavity patterns. How it happens is that the forward segments are cast in a fin-style regressive pattern that peaks at T+50. Then the aft segments are cast with a cylindrical-cavity grain that's progressive. The progressive thrust picks up steam at T+70. The sum of their thrust results in the desired profile. But don't think that varying combustion rates along the axis at different times doesn't pose some fun design challenges.
Here's a cross-section of the regressive forward segment of the space shuttle SRM grain.
https://www.pics4learning.com/details.php?img=srbcross-section.jpg Yes, a blank is put in there to reserve the cavity, and then the goopy "solid" propellant is poured in and allowed to harden. Then the blank is carefully removed. It's a lot trickier than the hold-out blank for the cylindrical castings.
SLS will use the same hybrid thrust profile, only with an additional booster segment. The regressive grain has been redesigned using better predictive models. This is not because SLS is as drag-sensitive as the shuttle in the same way. But the length and elasticity of the SLS stack means you need to temper your velocity through max-Q in case a lot of steering has to happen. Turning the rocket sideways to the slipstream at maximum aerodynamic pressure is risky.
As far as the internal structural geometry goes, an SRB is nothing like a liquid-fueled motor. The latter almost always integrates the thrust chamber and nozzle so that they are the same manufactured part. In the classic Rocketdyne method, both are made up of tubes furnace-brazed together into a single assembly with very smooth inner walls. Solid rocket boosters are almost always built as a motor casing with the other stuff attached later. This is because of how the casings have to be made. Many are filament-wound. The shuttle SRBs had the design constraint that the casing segments had to be modular. None of that allows for a nozzle to be manufactured integrally to the casing. Further, the nozzles are made from entirely different materials, usually because they have to be ablatively cooled. And there's the gimballing hardware, which has to go somewhere there. So where the nozzle and skirt assemblies have to be mated to the casing, there's a lot of geometry to create joinery strong enough to keep the nozzle end of the booster from being blown off under pressure.
Okay, this was about vibration. So let's get back to it.
Liquid-fuel engines suffer from three broad categories of vibration, in increasing order of (acoustic) frequency: chugging, buzzing, and screaming. Chugging is mostly caused by elasticity in the whole fuel piping system and rocket structures being excited by hydraulic fuel-flow and flight loads. That causes combustion instability by varying propellant flow rate. Pogo and chugging are the same thing. Buzzing is usually a localized problem in the propellant feed system, but same effect on fuel flow. Screaming is usually a harmonic resonance problem in the thrust chamber itself. It will occur even when propellant flow is stable. What these mostly have in common is the fact that the propellant is a fluid and is subject to many effects on its pressure and flow rate. They arise from the need for pumps, tanks, pipes, and sometimes some innovative structural designs to hold them all.
Solid-fuel motors have fluid-flow problems, just in a different regime of the design and with a different fluid. Yes, it's all a problem of fluids moving through a "pipe," but in the solid-fuel case the pipe is the entire rocket casing and the fluid is a torrent of combustion products driven by thermodynamics to do perplexing things. Channel flow is a fun problem even before you add rocketry. We can solve instability in propellant-feed systems by largely brute-and-klunky means like adding accumulators or thickening pipe walls. Mitigating flow instability in solid-fuel motors is a whole-system problem and as such has a plethora of additional constraints. (Don't get me started on the Ares-1 and those damn shock absorbers.) Hence we do what we can, but we mostly just live with it. As long as it stays below the acoustical loading limits of the payload, we just say it's going to be a bumpy ride as long as you ride SRBs. Northrop-Grumman just has to stay below the Orion capsule's shake-apart limit.
Stepping back several steps, the broad model that covers both rocket species (and, in fact, any fluid-management engineering) is the acoustic-mechanical coupling. Fluids flow. As they flow, they encounter solid-ish surfaces and impart loads on them. The solid parts behave according to ordinary mechanical harmonics. That in turn imparts loads back into the fluids, which then react. It's a feedback system. In good designs, feedback damps out. In bad designs, it doesn't. In acceptable designs if may reach an tlerable equilibrium. In all cases it requires attention both to fluid flow and to mechanical resonance. But when you draw the diagram of what affects what, the diagram is different for solid-fuel motors than for liquid-fuel motors. The phenomenon is there in all cases, but the boxes and arrows are different. When you see Elon Musk's rocket nozzles fluctuating under exhaust flow loads, that's an acoustic-mechanical coupling that's no different in its basic mechanics principles than obstacle vortex shedding between the forward and middle segments of the SRB. They're at the same time very different phenomena but also examples of the same general problem.
