  By Jim Davis    2021-04-05

# Coordinated Turns

Turns are one of the four fundamental helicopter flight conditions, alongside straight-and-level flight, climbs and descents. Pilots typically aim for “coordinated turns,” meaning turns with no lateral acceleration. The pilot will feel no left or right acceleration, and will be pushed straight down in her / his seat, just like in level flight. If you’re familiar with fixed wing flight, the high-level concept is the same. The pilot has a “slip ball” in the cockpit that he / she tries to keep centered throughout the turn.

You may also want to check out our coordinated turn calculator.

## Skid and Slip

The terms “skid” and “slip” refer to deviations from a coordinated turn. Skidding refers to going outside the intended circular arc. It occurs with excess pedal or insufficient lateral cyclic. For a right turn this means the nose is turning right too fast relative to the flight path and results in left sideslip (wind in the pilot’s left ear). The slip ball will be offset to the left, telling the pilot to apply left pedal, right cyclic or both. In this condition, we say the pilot is “skidding outside the turn.”

Slip occurs in the opposite sense. For a right turn, the flight path curls to the right faster than the nose turns right, resulting in a right sideslip. This occurs with excess right cyclic or insufficient right pedal. The slip ball will be offset right, telling the pilot to apply right pedal, left cyclic or both. In this condition, we say the pilot is “slipping inside the turn.”

From above, you can see why pilots are instructed to “step on the ball” to coordinate a turn. When the ball is displaced right, pressing right pedal will correct it. When it’s displaced left, pressing left pedal corrects it. This is a universal truth for both right and left turns.

## Thrust in a Turn

In a turn, the main rotor must produce more thrust than in straight-and-level flight. The rotor must provide the centrifugal force for the curved flight path while still supporting the weight of the helicopter and propelling it at the desired speed. This excess thrust can be obtained by increased descent rate or increased collective control.

## Single Control Turns

“Single control” turns with only lateral cyclic or only pedal are technically possible. However, a steady, coordinated turn typically requires a combination of both. Without pedal, a lateral cyclic input will roll the helicopter with excess lateral acceleration and slip. Directional stability will eventually provide yaw in the desired direction, but not enough to eliminate all slip.

## Math of a Coordinated Turn With some trigonometry, the math of a coordinated turn may be understood. The central idea is that the slip ball must be centered, so the pilot feels no sideward acceleration. The apparent acceleration due to gravity in the lateral direction must be precisely opposed by centrifugal acceleration to maintain the turn. The component of gravitational acceleration lateral to the fuselage is $$G\sin \phi$$ where $$G$$ is the total acceleration due to gravity and $$\phi$$ is the roll angle. The amount of centrifugal acceleration in the opposite direction is $$CF\cos \phi$$, where $$CF$$ is the total centrifugal acceleration: $$CF = v^2/R$$ where $$v$$ is the speed of the helicopter and $$R$$ is the radius of the turn. Equating these two accelerations gives

$$\begin{equation} G\sin\phi = \frac{v^2}{R}\cos \phi . \end{equation}$$

Using the fact that $$\tan\phi = \sin\phi / \cos\phi$$, this equation can be rearranged to

$$\begin{equation} G\tan\phi = \frac{v^2}{R}. \label{eq:turneq} \end{equation}$$

The vertical $$G$$s pulled by the helicopter are due to the vertical components of gravity and CF, namely $$G\cos\phi + v^2/R \sin\phi$$. Replacing $$v^2/R$$ using Equation \eqref{eq:turneq}, this becomes $$G(\cos\phi + \tan\phi \sin\phi )$$. Using the fact that $$\tan\phi =\sin\phi / \cos\phi$$ gives $$G(\cos\phi +\sin^2\phi /\cos\phi ) =G\frac{\cos^2\phi+\sin^2\phi}{\cos\phi} = G/\cos\phi$$. Hence, the vertical acceleration increases with $$1/\cos\phi$$ as shown in the plot below. ### Turn Rate

The turn rate $$\dot{\psi}$$, or rate of heading change, may be calculated as $$2\pi$$ divided by the amount of time it takes to complete a full circle. The latter may be computed as the circumference of the circle divided by the flight speed: $$2\pi R/v$$. Putting these together gives $$\dot{\psi} = v/R$$ and using Equation \eqref{eq:turneq} for R gives

$$\begin{equation} \dot{\psi} = G\tan\phi / v . \end{equation}$$

A standard rate turn is 3deg per second, which means a 180deg turn takes one minute.

Although the Euler roll and pitch may be constant in a turn, there will generally be nonzero body angular rates $$p, q, r$$ which may be found by transforming the Euler rate $$\dot{\psi}$$ into the body reference frame (often done with a rotation matrix). The results are below.

$$\begin{equation} p = -\dot{\psi} \sin \theta . \end{equation}$$

$$\begin{equation} q = \dot{\psi} \sin \phi \cos \theta . \end{equation}$$

$$\begin{equation} r = \dot{\psi} \cos \phi \cos \theta . \end{equation}$$

We skipped a technical detail above when discussing cyclic only turns. Upon entering such a turn, the helicopter may initially yaw in the wrong direction. This phenomenon is known as adverse yaw, and also occurs in fixed wing aircraft. In a right turn, the downward movement of main rotor blades on the right side of the aircraft increases their angle of attack. This tilts the lift vector forward, reducing torque. For American helicopters, with main rotors spinning counter-clockwise (viewed from above), engine torque yaws the nose right. The reduction in torque therefore yaws the nose left, opposite of what is desirable. This effect only exists with a roll rate; it's not present when holding a steady roll angle. After some sideslip develops, directional stability will overpower this phenomenon, yawing the nose back to the right.

## Trim Conditions

In this section, we examine how various values (flapping, pitch, torque, …) change with the roll angle/turn radius in a coordinated turn. We consider a set of “trim” conditions, all with zero climb rate and 100kts forward airspeed. We use the same aircraft weight and CG location throughout. Please note that this analysis is based on a specific helicopter model—many values (and even trends in rare cases) can be helicopter model dependent.

### Vertical load, collective, and torque

Plots of vertical load and main rotor thrust are provided below. They follow directly from the discussion above, increasing with bank angle to counter gravity and the increasing centrifugal force to maintain the tighter turn radius associated with larger bank angles. As expected, a substantial increase in collective and torque are required to satisfy the increasing thrust requirement.    ### Pedal and sideslip

As main rotor torque increases, tail rotor thrust must also increase to maintain zero net yaw moment. As usual, this is achieved with left (decreased) pedal input. But wait, this is a right turn! Of course, a pilot initiates a right turn with right cyclic/right pedal. However, if she wants to maintain altitude, she must increase collective as shown above and consequently end up with this (possibly counter-intuitive) left pedal offset. In helicopters equipped with an AFCS, the pedal displacement may indeed be right due to AFCS actuator inputs. In this case, the swashplate position is still the equivalent of a left pedal input.

The left pedal offset and associated tail rotor thrust push the helicopter right (in addition to adding a nose left yaw moment). There’s also a component of gravity $$GW\sin\phi$$ pushing to the right in the helicopter’s frame of reference. These forces cause the helicopter to slip to the right as shown in the sideslip plot below. (This sideslip can be removed with pedal and/or lateral cyclic, but results in a slight ball offset.)  ### Pitch rate, longitudinal cyclic and lateral flapping

Although the Euler pitch angle $$\theta$$ is constant, a coordinated turn requires a positive body pitch rate $$q$$. From the helicopter’s point of view, the aircraft is pitching up continuously as it travels around a circular arc in a coordinated turn. The sharper the turn (or the larger the speed), the larger the pitch rate, as shown in the plot below. A slight increase in main rotor flapping (“nose up”) facilitates this pitch rate (also shown in the plots below). The longitudinal cyclic increases as well, which would normally reduce flapping (“nose down”). The caveat here is that the increased collective (previous subsection) naturally induces more than enough positive flapping, so the cyclic is used to partially counter this “flap back.”   Back to home