Vmca factors
A core part of every multiengine curriculum is a good understanding of Vmca, or minimum control speed in the air. Per FAR 23.2135, "Vmc is the calibrated airspeed at which, following the sudden critical loss of thrust, it is possible to maintain control of the airplane. For multiengine airplanes, the applicant must determine VMC, if applicable, for the most critical configurations used in takeoff and landing operations." In many multiengine airplanes, this is denoted by a red radial line on the airspeed indicator.
An easy way to think of Vmca is a balanced state of tug-of-war between different torques acting on the center of gravity of the airplane. Torque is just arm * force, so we can look at how adjusting the forces and arms balances our torques. In the sketch below, we see the dead engine creating drag (dark red), the live engine creating thrust (green), and the rudder (purple) and wing/stabilizer (cyan) fighting those torques in the opposite direction. When we get too slow, the lift from the rudder and wings is no longer able to fight the engine and we lose control.
An easy way to think of Vmca is a balanced state of tug-of-war between different torques acting on the center of gravity of the airplane. Torque is just arm * force, so we can look at how adjusting the forces and arms balances our torques. In the sketch below, we see the dead engine creating drag (dark red), the live engine creating thrust (green), and the rudder (purple) and wing/stabilizer (cyan) fighting those torques in the opposite direction. When we get too slow, the lift from the rudder and wings is no longer able to fight the engine and we lose control.
When we look at the aircraft from the back in a banked state, you see the total lift vectors of the wing and stabilizers (orange), and then a decomposition into the vertical (light green) and horizontal (dark blue for the wing, cyan for the stabilizer) components. The horizontal components of lift create a net torque around the yaw axis that we see represented in the top diagram as the cyan arrow.
Now that we've got the basics down, let's play with the variables to understand how they make Vmca go up or down. I like to use the mnemonic MAD B WED (more on why here) to keep track of how these work on typical small twin trainers. A lot of these adjustments involve the basic lift equation and can easily be approximated, while others are a bit trickier.
Mass: the higher the mass of the aircraft, the more lift is required to fly. For a given bank angle, the total lift component, and hence the horizontal component of lift, HCL, will have to increase as the mass does, thus giving us more of that friendly cyan torque. A linear mass increase will give us a linear lift increase, which at the same bank angle gives us a linear HCL increase, and consequently a linear HCL-induced torque moment change. We know that the rudder acts as an airfoil and follows L= ½ CL * S * ρ * α * v^2, so to keep the cyan+blue vectors the same combined length, we can afford a linear decrease in rudder lift. A linear decrease in lift means a square-root change in speed. Thus, we get an inverse square-root scaling Vmca=1/√∆w, ceteris paribus. One thing to be careful with on checkrides is the assertion that "More mass is an inherent good because it increases the yaw inertia of the aircraft." There are two reasons to avoid this claim:
Mass: the higher the mass of the aircraft, the more lift is required to fly. For a given bank angle, the total lift component, and hence the horizontal component of lift, HCL, will have to increase as the mass does, thus giving us more of that friendly cyan torque. A linear mass increase will give us a linear lift increase, which at the same bank angle gives us a linear HCL increase, and consequently a linear HCL-induced torque moment change. We know that the rudder acts as an airfoil and follows L= ½ CL * S * ρ * α * v^2, so to keep the cyan+blue vectors the same combined length, we can afford a linear decrease in rudder lift. A linear decrease in lift means a square-root change in speed. Thus, we get an inverse square-root scaling Vmca=1/√∆w, ceteris paribus. One thing to be careful with on checkrides is the assertion that "More mass is an inherent good because it increases the yaw inertia of the aircraft." There are two reasons to avoid this claim:
- We measure Vmca as the speed at which we lose the torque tug-of-war with the engine, not the rate at which it loses directional control. The yaw inertia influences the rate, not the speed at which it occurs. On the checkride, we're supposed to recover at the first loss of DC, regardless of whether it occurs briskly or gradually.
- More mass doesn't necessarily mean more yaw inertia. The mass, as well as the distribution thereof, jointly determine that. If you put two people in a Seminole and fill the saddle tanks to get 3700#, the mass of the fuel is pretty far out there. Conversely, if you cluster 3 or 4 people in the cabin near the CG with mostly empty tanks and a 3701# gross weight, you'll still have less yaw inertia because the moment arms to the masses are shorter. You can observe a figure skater illustrating this, or spin around on a swivel chair and pull your arms and legs out, then extend again, to observe the preservation of angular momentum.
Arm: as the CG moves forward and aft, the lever arm to the rudder and stabilizer changes as well, thus causing a linear change in torque. For the rudder at least, this means that for a linear reduction in arm, we get a square-root increase in Vmca. The HCL from the wing and stabilizer are a bit trickier to model because there's more going on. With an aft CG, the wing HCL will be slightly smaller, but pull closer to the CG. Meanwhile, the stabilizer will not be creating as much trim drag to fight the forward CG, so its total lift vector will be shorter, and the HCL will suffer as well. As with the rudder, the lever arm will get shorter, so we lose torque from both variables.
Performance is better with an aft CG, due to lower trim drag to balance it all out.
Drag Devices: the basic idea here is that hanging gear, cowl flaps, and wing flaps into the propwash will create a little bit of drag in the decelerated slipsteam on the dead side (short dark blue arrow), and create a lot of drag in the accelerated slipstream on the good side (long dark blue arrow). This asymmetry in drag creates a yawing torque against the pull of the good engine, thus stabilizing the aircraft.
Performance is better with an aft CG, due to lower trim drag to balance it all out.
Drag Devices: the basic idea here is that hanging gear, cowl flaps, and wing flaps into the propwash will create a little bit of drag in the decelerated slipsteam on the dead side (short dark blue arrow), and create a lot of drag in the accelerated slipstream on the good side (long dark blue arrow). This asymmetry in drag creates a yawing torque against the pull of the good engine, thus stabilizing the aircraft.
There are a couple of subtleties to keep in mind with this though. Some flap systems, like that on the Seminole, provide mostly drag and very little incremental lift. Clean stall is 57 KIAS while dirty stall is 55 KIAS. Sqrt(57/55)=1.018, so our lift coefficient from the flaps only improves by 2% while drag shoots up. There might be flap designs out there that produce much more lift with minimal incremental drag, so you could suffer from more destabilizing lift on the good side and not much drag to counteract it. Thus, there might be an airplane out there (someone please send me an example if able) where a low flap setting is worst for Vmca, and the clean and full flap combinations have a better one. Either way, be careful making blanket assertions about flaps always improving Vmca if you get asked on a checkride.
The keel effect: there's a saying that landing gear acts like the keel on a sailboat, providing both a lower CG as well as aerodynamic help in stabilizing the aircraft's directional control. Most gear legs I'm aware of are cylindrical and do not have well streamlined flow, in contrast to a keel fin, so they probably don't help very much on the aerodynamic front. Nose gear legs and doors, being ahead of the CG, add drag up front, which actually destabilize the aircraft. As far as longitudinal CG goes, a forward-extending nose leg does move the CG forward, but the effect is trivial (I thought I saw a reference in the PA44 POH confirming that, now I can't find it). Likewise, the vertical CG shift from lowering the wheels is not that big either. Napkin math: let's say the gear system on a Seminole weighs 100 lb, and the extension puts its CG about 20 inches lower than when stored. Applying the weight shift formula 100/3800=CG/38, we get about a half-inch vertical shift. How much does that help us? Not sure, but it's probably not the strongest factor. I'd wager that the majority of the help from the gear comes from the asymmetric drag, not the keel effect. Again, when discussing this on a checkride, make sure you understand the nuances.
Bank: As we bank further into the dead engine, we get a larger HCL from both the wing and stabilizer, which give us favorable torque around our CG that helps us stabilize the aircraft.
The keel effect: there's a saying that landing gear acts like the keel on a sailboat, providing both a lower CG as well as aerodynamic help in stabilizing the aircraft's directional control. Most gear legs I'm aware of are cylindrical and do not have well streamlined flow, in contrast to a keel fin, so they probably don't help very much on the aerodynamic front. Nose gear legs and doors, being ahead of the CG, add drag up front, which actually destabilize the aircraft. As far as longitudinal CG goes, a forward-extending nose leg does move the CG forward, but the effect is trivial (I thought I saw a reference in the PA44 POH confirming that, now I can't find it). Likewise, the vertical CG shift from lowering the wheels is not that big either. Napkin math: let's say the gear system on a Seminole weighs 100 lb, and the extension puts its CG about 20 inches lower than when stored. Applying the weight shift formula 100/3800=CG/38, we get about a half-inch vertical shift. How much does that help us? Not sure, but it's probably not the strongest factor. I'd wager that the majority of the help from the gear comes from the asymmetric drag, not the keel effect. Again, when discussing this on a checkride, make sure you understand the nuances.
Bank: As we bank further into the dead engine, we get a larger HCL from both the wing and stabilizer, which give us favorable torque around our CG that helps us stabilize the aircraft.
In the sketch above, the aircraft at right in a steeper bank has longer blue and cyan arrows. HCL=sin(φ)*L, where φ (the Greek letter "phi") is bank angle and L is the total lift vector. For small angles, we can approximate ∆sin(φ)=∆φ, or basically a linear change in bank gives a linear change in HCL. It's the same "close enough" philosophy we use with the 60:1 rule. As we all know from practicing steep turns, our VCL decreases in a steeper bank, VCL=cos(φ)*L. Thus, there's a practical limit on how much bank we can use to regain control before we lose too much vertical performance. This is why 23.149 (since superseded by 23.2135) gave a limit of φ=5* and the ACS tells us on the VMC Demo task that we're not allowed to bank past 5* (CA.X.B.S3). If you want to play with it, go ahead and do a VMC demo at a safe altitude and bank to 6-10 degrees. You can hold onto your heading a little longer before you lose control and need to recover. More bank is more control, just keep in mind that φ=5* is a testing and certification limit.
Modeling the effect of bank on Vmca is a bit tricky because most aircraft are probably tested with φ=5*, so it's hard to back out the effect of the rudder and the HCL and then multiply the HCL by ∆φ.
Performance effects are tricky to model as well. If you fight a dead engine with rudder only, then the leftward pull of the rudder and the forward pull of the right engine will result in a net forward and left motion as shown by the pink arrow in the sketch below.
Modeling the effect of bank on Vmca is a bit tricky because most aircraft are probably tested with φ=5*, so it's hard to back out the effect of the rudder and the HCL and then multiply the HCL by ∆φ.
Performance effects are tricky to model as well. If you fight a dead engine with rudder only, then the leftward pull of the rudder and the forward pull of the right engine will result in a net forward and left motion as shown by the pink arrow in the sketch below.
If you're a tactile learner, put a model plane on a table and push forward on the right engine and push left on the rudder to maintain heading. The model will move diagonally forward like the pink arrow sketch. As we all know from being barked at by our instructors for not using enough inside rudder, HCLs without rudder input add a vector to the inside of a turn, which we can use to our advantage in this case. If we add the cyan and dark blue HCL arrows back in from the earlier sketches, it pulls the pink total movement vector into a forward line aligned with the aircraft centerline:
If you're still pushing that model around your desk, go ahead and have a buddy push the engine while you nudge the rudder left and the wing right. If you balance them, the model should track straight along the table.
From a total performance standpoint, the drag reduction from flying straight is worth a small loss of VCL, which is why the best performance value for φ is 2-3* on most types.
Windmilling critical engine: when you have a critical engine windmilling, you're basically using free stream air to spin your engine, which creates a lot of drag. If you've ever used engine braking on a standard transmission car to slow down, it's basically the same process of using the vehicle's inertia to spin the engine through its compression and exhaust cycles. This creates a yawing motion (red arrow). On aircraft with wing-mounted engines, it also creates a rolling motion. As the free stream air is slowed by the windmilling prop, it creates a decelerated slipstream which results in less lift created on the dead side. The reduction of lift on the dead side, when combined by the propwash-induced lift on the good side, creates a rolling moment towards the dead engine (shorter lift arrows in lower sketch):
From a total performance standpoint, the drag reduction from flying straight is worth a small loss of VCL, which is why the best performance value for φ is 2-3* on most types.
Windmilling critical engine: when you have a critical engine windmilling, you're basically using free stream air to spin your engine, which creates a lot of drag. If you've ever used engine braking on a standard transmission car to slow down, it's basically the same process of using the vehicle's inertia to spin the engine through its compression and exhaust cycles. This creates a yawing motion (red arrow). On aircraft with wing-mounted engines, it also creates a rolling motion. As the free stream air is slowed by the windmilling prop, it creates a decelerated slipstream which results in less lift created on the dead side. The reduction of lift on the dead side, when combined by the propwash-induced lift on the good side, creates a rolling moment towards the dead engine (shorter lift arrows in lower sketch):
Neither of these effects is good for performance or controllability, which is why we need to feather the dead engine ASAP. Modeling the effect of the windmilling and its interaction with drag devices is beyond simple thumb rules, so I'll leave that to the aerodynamicists.
Engine Power/Thrust: The stronger the good engine pulls, the more torque you get to destabilize you. If you're holding bank constant, then your HCL and rudder lift should scale quadratically with speed, so you should get a square-root scaling Vmca=√∆T.
Density: Less dense air typically results in less power and thrust from your good engine, which results in less adverse torque and thus lower Vmca. If you have an airplane where thrust scales pretty linearly with density, you can just say T=ρ and substitute the above equation and say Vmca=√∆ρ. There are a few things to keep in mind with this:
Engine Power/Thrust: The stronger the good engine pulls, the more torque you get to destabilize you. If you're holding bank constant, then your HCL and rudder lift should scale quadratically with speed, so you should get a square-root scaling Vmca=√∆T.
Density: Less dense air typically results in less power and thrust from your good engine, which results in less adverse torque and thus lower Vmca. If you have an airplane where thrust scales pretty linearly with density, you can just say T=ρ and substitute the above equation and say Vmca=√∆ρ. There are a few things to keep in mind with this:
- Vmca, as defined above, is the calibrated airspeed at which we lose control. This means that while less-dense air will result in a performance penalty at the same TAS, we're using CAS, which already compensates for density. Don't let that trip you up if someone asks "Do the wing and rudder get less effective with less density?"
- Boosted/flat-rated engines: Turbocharged or turbonormalized reciprocating engines will often produce rated power up to an altitude where the wastegate is fully closed (critical altitude), at which point you'll start to get a drop in output. Some turbines have a torque limiter that likewise limits power below a given altitude (flat rating). This means that below the critical altitude, Vmca will stay almost constant as you climb. You'll still get a slight drop in thrust: as the air thins, you need more overspeed (F=ma) for the same thrust. Accelerating the air costs you KE=1/2mv^2, so if m drops, v^2 needs to pick up the slack and you run out of energy sooner. More on that here.
CHallenges with "SMACFUM"
There's a mnemonic out there that's based on 23.149, the original certification standard, that's commonly taught:
S: Standard day
M: Most unfavorable weight
A: Aft CG
C: Critical engine windmilling
F: Flaps and Gear out
U: Up to 5 degrees bank
M: Max power
In my experience, the literal conditions from the cert standard make the variables easy to memorize, but harder to understand:
S: Standard day
M: Most unfavorable weight
A: Aft CG
C: Critical engine windmilling
F: Flaps and Gear out
U: Up to 5 degrees bank
M: Max power
In my experience, the literal conditions from the cert standard make the variables easy to memorize, but harder to understand:
- "Standard day" may be the test condition, but it doesn't prompt you to consider worse-than-standard conditions. When I ask students on stage checks "Will Vmca be better or worse at 30.10 and -10 degrees," many of them freeze. Having "density" as a noun forces you to consider what the change in density does. Lots of students likewise get tripped up by the density effects on the airfoils and don't keep TAS and CAS separated.
- "Up to 5 degrees bank." When I hear that regurgitated, I ask "Is controllability better at 5 degrees or 7?" Many students don't have a good answer. Teaching that an increase in bank leads to better controllability due to favorable HCL vectors helps break this mold. Start with the theory, then discuss the cert standard and ACS. The jingle I like is "Keep phi under 5."
- Personal preference: I find that adjectives are harder to remember than the active nouns (the 2 Ms and the U). That's why day 2 of multi training I worked with my instructor to come up with MAD B WED.