Is trim linear with speed?
 

Question in my head that came from the A320 stall thread....

In general is trim change linear with speed change?

And more specifically; is it linear with FBW aircraft - A320?

My thoughts would be that according to the lift formula you’d would require proportionally more nose up trim for every knot of speed loss.

It has to be correlated but if it’s not linear what is it?

Thanks in advance.

Rearranging the lift equation, CL varies by the inverse of the square of speed (or, speed varies by the inverse square root of CL), so no. There are further relationships to consider (CL to AOA, and AOA to stab angle) which, even if they are linear, there's already a square link in the way.

Put simply, as speed increases, every further knot is a smaller decrease of AOA. Conversely as AOA decreases, every next degree of AOA decrease, is a bigger speed gain. Or, as speed decreases, every next knot is a smaller AOA decrease. That's why on final in rough air, a ham fisted pilot can saw the yoke nearly from stop to stop and it's little more than an annoyance; while in cruise, that would rip the wings off.

Minor correction - it's not the CL (Coefficient of Lift) that changes. The CL stays fairly constant with speed (until you approach the speed of sound and compressibility comes into play). It's the actual lift component that varies with the square of airspeed.

Are you talking about 1G, straight line flight? If so, I disagree. Actual lift stays constant (it has to equal weight, which is constant) throughout all speeds. So CL varies inversely with speed. As long as we're not close to the stall, CL essentially means AOA from the pilot's perspective.

MAXIMUM CL is fixed (as is AOA) and MAXIMUM lift varies. But these maximum values define the limit of the V-G diagram (to the left of Va), not straightline flight.

CL is a function of angle of attack. It's not a function of speed (again, below the transonic region). You don't see CL plotted vs. speed - it's vs. AOA. Yes, in level flight, lift stays constant - pretty much by definition. But if you speed up, the CL doesn't change, the AOA does - which moves the CL to a different point on the CL/AOA curve.
CL is a dimensionless term that measures the ability of a particular body shape to generate lift. You combine that with density and velocity squared term to determine the amount of lift.

"CL is a function of angle of attack." - yes. And as long as we're not near the stall, and it's a high aspect ratio, low sweep wing, a pretty simple function at that. Say, it looks something like this. This wing stalls (say, the red Reynolds) at 11 degrees AOA and 1.2 CL. Below 9 degrees, it's a linear relationship of 0.1 CL per degree AOA. Multiply or divide one by 10, you get the other. This is what I meant by saying that from the pilot's perspective, they're essentially synonymous (except near the stall). Just as you can think of the elevator as the AOA control, you can think of the elevator as the CL control. Pull back and you increase the CL - that is, until shortly before the stall. (Then, it flattens out, and after that, starts decreasing).
https://cimg0.ibsrv.net/gimg/pprune....632cad3629.png
"You don't see CL plotted vs. speed - it's vs. AOA." Also true.

But,

"It's not a function of speed." - not necessarily true. If we constrain to a 1G (or any constant G) condition, AOA becomes a function of speed, so therefore, CL is then also a function of speed.

"Yes, in level flight, lift stays constant - pretty much by definition. But if you speed up, the CL doesn't change, the AOA does - which moves the CL to a different point on the CL/AOA curve."

First you say the CL doesn't change, (false) but then you say it moves to a different point - which is true, and which means it changed. Let's say on the above graph we start stable at some mid condition, at 5 degrees AOA, and 0.7 CL. And, filling out some constants, this means 100 knots. If we move the stick slightly forward, let everything stabilize, the AOA will decrease - say by one degree so now we're at 4 degrees AOA, and the CL has changed to 0.6. And, instead of 100 knots, now this yields an increase to 110 knots.

"CL is a dimensionless term that measures the ability of a particular body shape to generate lift." What I'm getting is that you're using CL to mean maximum CL, or CLmax. CLmax, yes, shows the "ability" to make lift. The liftier the wing (high camber, high thickness, etc.) the higher the CLmax is. But the CL can be (and, most of the time, is) at any value below the maximum, as dictated by the current conditions (and commanded by the stab and elevator). I.e., CL represents the current lift, not the ability of maximum lift (that would be CLmax). If you take the lift equation and cross out all the constants (for straightline flight, in a given situation), it's plain that CL varies inversely with speed.

L = CL * 0.5 * density * speed^2 * area

Here's a representative graph I really like, from Aerodynamics for Naval Aviators, with the steady state speeds overlaid (at the risk of introducing a whole 'nother set of example numbers to confound future posts). Really wrapping everything together, and back to the trim relationship question at the beginning of the thread. You can see how at the faster speeds, every degree of AOA yields a larger speed difference than at the lower speeds.
https://cimg4.ibsrv.net/gimg/pprune....43d7078bea.jpg

I think we're arguing minutia here, but it's misleading to say that CL changes with speed - it doesn't. If you keep the same configuration, and you speed up - you'll create more lift (as a function of V squared) and you'll climb. If you're talking 'straight and level', it's not the CL that changes with speed change - it's the flight profile that changes (specifically the AOA), which in turn moves the CL to a different point on the AOA curve.
Yes, lift is a function of speed squared - but not because the CL changes. It's because the dynamic pressure is a function of speed squared, and lift is linear with dynamic pressure.

What is the exact reason for your question?
Leaving the protections of A320 aside, the effect of speed on trim isn't different to a let's say B737.

Yes, some of what we're talking about is minutiae, and we're talking past each other. In my paragraph where I distinguished CL (current) from CLmax, I offered my impression of what the disconnect is. I'm still not sure what your thoughts on that are, and that would be helpful.

Here, I'm seeing another potential disconnect, which is that it seems you're objecting to the direction of causality - is it the speed change that drives the CL change, or the CL change that drives the speed change? Well, ultimately, it's not that important, and may work both ways in different real-life scenarios. But, kind of like the question of whether the Earth pulls on the Moon or the Moon pulls on the Earth, who cares? Richard Feynman's "shut up and calculate" gives the answer of how the objects will behave.

And the answer is that CL is absolutely, positively correlated with speed. From the most recent attachment, when that plane is flying at stall speed (100 knots), it's flying at 1.50 CL. At 110 knots, CL = 1.25, at 120 it's 1.05, etc.

If you command 12.5 deg. AOA with the elevator, which is the same as saying you're commanding a CL of 1.05, the steady state speed will be 120, period dot.

You gave some examples, but I'm afraid were not careful and mixed steady-state with dynamic conditions:

You probably meant to say "throttle up." From that, you'll speed up only temporarily and from that climb (lift exceeds weight, CL is unchanged) - until you reach the climb angle (after the phugoid has damped) where the new thrust is equaled by the drag combined with rearward component of weight, and lift equals weight (I think we can agree to ignore the cosine of climb angle tiny reduction) and is therefore unchanged - so Lift, CL, AOA, and speed are all at their original values.

Short version of the above: if you just throttle up, ultimately you won't speed up, you'll climb. If you want to speed up permanently, that can not be achieved without reducing CL.

I think the OP is looking at long stab considerations?

If so, the general variation of stick force (which is the thing you might wish to trim out) with speed moving away from the trim speed is sort of linear but, normally, with a bit of roll off. The important thing is that it be acceptably monotonic.

If one runs a Google search one can find lots of relevant curves.

You can answer this question yourself by trying it in ANY airplane or simulator! At low speed, you can move the trim wheel or hold the button for a second or 2 to make a desired change. At high speed you only need a blip of the button or a minor movement of the wheel.

Others have given you the formulas...

john_tullamarine;

You're right, everything I've been saying has been about the main wing AOA. This part I'm less sure of than my other posts, but: I think as a first approximation, it's fair to say that the trimmed main wing AOA probably changes linearly with stab deflection. After all, the stab is a wing like any other, subject to the same linear lift-as-a-function-of-AOA, then forcing the tail up or down to rotate the main wing.

At lower speed, the change in stab deflection per speed is greater as we've said (to make up for the lower contribution to Lift from the speed^2) but by the same token, there is less force resisting that deflection. So the 2 effects more or less cancel each other out, so in terms of stick force, the trim per speed stays more or less constant (though in terms of stab deflection, or wheel-rolling, or switch-holding-down, the trim per speed changes)