What does stick fixed and stick free longitudinal stability mean in simple terms?

Question

What is the significance of stick fixed and stick free terms in aircrafts with mechanical control and modern fly-by-wire control?

Answer 1

Trim the aircraft. Do a "doublet" (means stick forward and back to the trimmed position for a short moment, or rudder deflected and neutral again) to upset the trimmed state.

Now watch the response while you keep all control surfaces at their trim position, making sure the control surfaces don't move when the aircraft goes through its motions. This is quite easy on GA aircraft and big aircraft with powered controls (if you do this on an Airbus, switch it to direct law).

Next, repeat the maneuver but let go of the stick respectively the pedals. Watch how the stick or pedal position moves with angle of attack respectively sideslip changes: This is your control surface floating freely. Again, watch the response. Normally, it will now take longer for the resulting oscillation to die down and the oscillation frequency should be slightly lower.

The first is the fixed-stick aircraft stability, the second the stick-free stability. Normally, stick-free stability is lower and proportional to the stick forces when flying away from the trim point. The difference between both is determined by the hinge moment coefficients of the control surface. For the elevator the floating angle $\eta_f$ is $$\eta_f = -\frac{c_{r0}+c_{r\alpha}\cdot\alpha}{c_{r\eta}}$$

with $c_{r\alpha}$ the hinge moment coefficient over angle of attack $\alpha$ and $c_{r\eta}$ the hinge moment coefficient over control deflection. Both coefficients are normally negative and the absolute value of $c_{r\eta}$ is larger that that of $c_{r\alpha}$. When $c_{r\alpha}$ is zero, both stabilities are equal because the floating angle is zero.

Another way to study stick-fixed pitch stability is to trim the aircraft at two different speeds or load factors and to measure the trim change necessary between trimmed state 1 and trimmed state 2. The result is the speed stability rsp. the maneuver stability.

Stick-free stability is only relevant in case of manual controls. FBW aircraft only exhibit stick-fixed stability except for the very unusual case that the control surface actuators are overpowered by the aerodynamic forces.

Answer 2

Consider an aircraft in stationary, horizontal, trimmed flight. Longitudinal stability could be described as the tendency for the aircraft to return to its original position after an external event has changed the Angle of Attack $α𝛂$.

• Static stability: with the pitch stick held at a fixed position, an aircraft is statically stable if the aircraft has a tendency to return to its original position after an attitude change by an external event. Without having to give a stick input. So a wind gust causes a nose up moment, changes the aircraft attitude, and the increased attitude causes an opposing aerodynamic moment that drives the aircraft back to its original position once the gust is gone. This can only be achieved if the centre of lift is aft of the centre of gravity:

  $CN_2$ > $CN_1$ when $α𝛂_2$ > $α𝛂_1$

• Dynamic stability. So our statically stable aircraft is driven back by aerodynamics to where it was, but what happens then? If it reaches neutral, then overshoots further to the other side, then back through neutral and further yet again, the response diverges and the aircraft is dynamically unstable. Not a good situation because in order to return to trimmed flight, we need to generate periodical stick inputs that counteract the natural response. We would like the aerodynamic forces to take care of this.

• But what happens if we're dynamically stable with fixed pitch stick but don't hold it in position during a periodic movement? Does the returning angular motion cause an elevator deflection that counteracts the motion, or one that amplifies it? That depends on the behaviour of the elevator after an external event has caused an increase in $α𝛂$. Does the stick start to wobble, just like the aircraft did, and what sort of wobble is it? One that returns to where it was or is it diverging? This is a function of the sign of the hinge moment of the elevator. If we have an elevator that floats up after an increased $α𝛂$, it will tend to increase α$𝛂$ even further. So that arrangement will reduce the static stability of the airframe.

The above is a nutshell description of aerodynamic stability. There is more to consider, like stick manoeuvre deflection and force stability, but that may make this post too lengthy. However, we can see that for an aerodynamically stable airframe, we need an aerodynamic moment that counteracts our motion. OK for a Cessna but not desirable for high manoeuvrability aircraft like fighters. For these, we want the aaeroforces to help us make the turns - but that leaves the pilot with an unstable equilibrium, like trying to balance on top of a big inflatable ball. Tiny stick inputs are constantly required to fly in a straight line, because aerodynamics constantly want to deviate us from the trimmed position.

And that is where fly-by-wire comes in. It uses a computer to generate the small inputs required to keep the airplane neutral - constant small deflections that are not noticeable to the pilot, he only experiences a stable aircraft. Give a stick input, and all aerodynamic forces cooperate to make a rapid manoeuvre.

I've heard that Formula 1 driver Michael Schumacher had his car tuned just like that: directionally unstable so able to turn quickly but requiring constant small corrections at the steering wheel. Not everybody's cup of tea.