When choosing a frame for a drone, something I hear mentioned occasionally is how flexible the carbon fiber is. It seems that the general opinion is: more flex in a frame is worse. How would having a frame with flex in it be bad for a high-powered quadcopter? Would it impact the efficiency and performance of the quadcopter?
Stiffness, oscillation, and vibration damping are separate concepts but all somewhat relevant to the question. Each affects flight dynamics, and each is (somewhat) a function of material choice.
For example, titanium is very stiff, and it rings like a bell. CF is very stiff, but it rings like mud. Aluminum is in between.
Stiffness is a good thing when you don't want your structure to be compliant[*]. Stiffness means that the frame isn't deflecting, which is important for the presumed laws of motion. Put another way, a quadcopter's autopilot assumes that all the motors are pointed in a fixed direction[**]. If those directions change, i.e. because of flexing, then the most fundamental assumptions of its flight dynamics are false. It's no surprise flight flight performance degrades!
Of course, any structure will bend/deflect/twist under load, no matter how stiff. What's important is whether it does any of these sufficiently enough to compromise your flight goal. If you're trying to do telescopic videography then a little bit of error is immediately visible in the results. If you're learning to fly, then so long as the autopilot keeps things sunny-side up you're fine.
Oscillation, such as control-surface flutter, occur when the entire structure starts deforming like a spring. This can dramatically build up and quickly rip the airframe to shreds. (It really is that fast, that violent, and that scary.)
If a quad's arms are insufficiently damped, oscillations can occur. Most frequently, it will occur as a twist about the arm axis, but I've also seen it happen in the flapping direction. Because of differences in dynamics between a multirotor and a fixed-wing, it's unlikely that your quad will suffer inflight failure but it's also very hard to keep control, both for you and the autopilot.
The closer you get to instability, the more energy you have to expend to keep things stable. In the case of a multi, at the very least this translates into decreased flight time.
The more powerful your input (e.g. a powerful quadcopter), the easier it is to drive a system past a critical damping threshold. The Tacoma Narrows Bridge was famous for this when the wind was blowing and the bridge was slightly wet. More recently, these out of control harmonic oscillations have famously been seen on the Millennium Bridge in London (surprisingly due to human footfall) and the Humen Perl River Bridge in China (heavy winds).
A full discussion of how to control oscillation is outside the scope of this answer. However, it is relevant to note that some materials are springier than others, and these materials are the ones more likely to have pathological oscillation.
Vibration is a form of oscillation, but I'm breaking it out so I can specifically address how it affects sensor data. Vibration is an issue when it causes signal integrity issues with the IMU (i.e. accelerometer + gyrometer)[***]. This can confuse the autopilot, leading it to make incorrect estimations of the aircraft's roll-pitch-yaw, and as a result respond inappropriately. This can and does cause crashes.
Again, a full discussion of how to control vibration is outside the scope of this answer. And once again, the springier the material is, the more it is likely to have pathological vibration.
Composites vs. metals
Composites (e.g. carbon fiber, fiberglass, kevlar, ...) have what are called anisotropic properties. This means that they don't have the same properties in all directions. Consider a square sheet of aluminum. No matter which axis you pull it on, it has the same strength, stiffness, heat transfer, etc... This is isotropic.
Now consider the same sheet made out of unidirectional carbon fiber. Along the carbon fiber axis it's incredibly strong. But turned 90 degrees, where the only thing holding it together is the glue, you'll find that it has no strength whatsoever.
You would be correct in assuming that this difference in properties extends to oscillation and vibration. The upshot is that composites can be tricky to chose because two seemingly similar structures (e.g. a tube) can have very different properties.
Yes, a semi-rigid frame will cause vibrations while flying because it alters the nature of motor and propeller oscillations as the frame parts move around. This will cause a drop in efficency as the flight controller attempts to combat these vibrations and return to stable flight, using slightly more power in the process.
Excessive vibrations in parts of the frame caused by semi-flexible pieces and or unsecured electronics parts can also cause damage to the frame and electrical components and increases the risk of accidents where the quad falls apart prematurely or drops out of the air.
Strictly speaking, any energy that goes into contorting the frame is energy that didn't go into accelerating/decelerating the quad, which is to say that bending the frame inflight will decrease the efficency of the whole system. This is also true about the thrust force vector, which will deviate slightly from vertical when the frame is bent, decreasing efficency. These losses would usually be very tiny and almost negligible because the amount of bending possible on most commercially available frames is miniscule. If these kinds of losses are a problem on your quadcopter, you need to address them before you fly because they will likely cause your quad to fall out of the sky and or cause damage to itself.
However, there are other issues with frames that allow parts to flex freely. Semi-rigid frames will move with the motor and propeller vibrations and bend when motor thrust changes, causing even more trouble for the flight controller and can impact flight performance. This can be another source of efficency loss as the flight controller tries to compensate for oscillations that are caused by parts of the frame vibrating uncontrollably. This is why it's always recommended to fully tighten screws and make sure all parts are carefully secured to the frame before flight.
I am going to jump in here with more than comments to other answers and I am going to answer this question in my own unique way. First, I have a similar related question, and I recently posed it to an industry expert. I asked a question about stiffness of Carbon Fiber frames. Sorta like this: "How stiff is stiff enough?"
And the answer which I will paraphrase here: "nobody measures the stiffness of multirotor frames". And... "They fly it till it breaks and then make it stronger/stiffer if needed." I took liberties with this Q&A so that nobody googles it to find the conversation. I believe the expert told me it did not matter. I was happy to get an answer from him.
So, the original question stands as asked. "How does the frame stiffness affect a quadcopter?" And my version: "How stiff is stiff enough?" I believe that no study has ever been performed on stiffness and there may never be one.
If you can bend it with your hands it is probably too flexible. When it breaks make it stronger.
If anyone has a source for test data of multirotor frame stiffness, I am eager to learn.
The Geeling Lightning 120 is known for its very flexible carbon, but when I fly it in my 100 foot back yard with many trees, I notice no problems. I love this 3" quad. I did however have enough damage to the props so that I was eventually getting Jello. No problem. I put new props on and the Jello was gone. I have not had the Jello problems with stiffer frames and damaged props.