Helicopter Flight DynamicsHelicopter Flight Dynamics

Chapter 3: Helicopter Motion Equations

Helicopter Motion Equations

Origin

For rigid body motion

: velocity of mass center : angular velocity of rigid body

: angular momentum

1. For the calculation, the equations can be written in scalar form. The left side of the equations are aerodynamic force and gravity.

2. The right side of the equations is the inertial force (moment) caused by (angular) acceleration.

3. In the force equation, the cross product of vectors is the centrifugal force. In the moment equation, the the cross product of vectors is the gyro moment.

Helicopter Motion Equations

Longitudinal Motion

is the summation of aerodynamic forces along the x axis.The three items in the right side are equivalent to ma. The first item comesfrom the acceleration, while the last two items are the inertial accelerationsproduced by the helicopter curve motions.

Helicopter Motion Equations

For the pitching motion, we have:

The lift side is the summation of aerodynamic moments. In the right side, the first item is the inertial moment caused by angular acceleration, the second item is the gyro moment, and the third item is effect of centripetal force.

Lateral/yaw Motion

Helicopter Motion Equations

The Relationship Between Angular Motions

The attitude angles are respect to gravity axes. While the angular velocities are measured in body axes.

The 6 equations of helicopter body and 3 angular motion equations consists of the governing equations of helicopter flight dynamics.

The relationship between angular velocities and helicopter attitudes rates can be written as:

ZD

Z1

Y1

X1

yz

zy

zyx tg

sincos

cos

sin

cos

cos

sincos

Summation

1. The governing equations of helicopter flight dynamics are nonlinear. 2. The motions of helicopter due to the perturbation (variations of

velocities or external forces) and controls (applied by pilot) are called as “stability”,and “control”. By using these equations we can analysis the helicopter stability and control.

3. The aerodynamic forces on left side of the equations comes from the contributions of main rotor, tail rotor, fuselage and empennage. They are the functions of control and motion variants in the right side.

4. Especially,The controls, induced velocity and flapping motion are implied in the items of main rotor and tail rotor aerodynamic forces. Therefore, the governing equations are very complicated.

tXXfX ,,

tUXXfX ,,,

Stability Analysis

Control Analysis

Helicopter Motion Equations

Linearization of Helicopter Flight Dynamic Equations

Motivations

1. To simplify the analysis and calculation, change non-linear differential equations into linear differential equations

2. To make sense the physical meanings3. To analysis the effect of parameters on helicopter motions and guide

the design

Feasibility

For the start time of external perturbation, the variation of motion is small with respect to the reference steady state condition (trim condition) and can be treated as linear variation approximately.

Helicopter Motion Equations

Assumptions:

1. Assuming small angles: attitude angles, attack angle, flapping angles are all assumed small so that the cos()=1, sin()=0.

3. Neglecting the items of more than second order

2. Assuming small perturbations: 1) The variations of (angular) velocities are small relative to the refe

rence steady state value.

2) The variations of forces are small so that:

Helicopter Motion Equations

Longitudinal Motion Equations in Hover

Lateral/yaw Motion Equations in Hover

1. The right side of equations contains control derivatives. While the left side contains the aerodynamic derivatives (stability and damping derivatives).

2. We can obtain the equations of studying helicopter stability by setting the right side is equal to zero (no controls applied).

3. We can obtain the equations of studying helicopter response to given controls.