The Effect of Winds

The Effect of Winds

The equations of motion Vinh presents for a flight vehicle are 3 kinematic (position), 3 force, and 1 mass equation. Depending on what effect is to be modeled, more equations will be added.

Wind on a Point Mass

Only forces can act on a point mass. Therefore additional equations are not required. However, some of the existing equations need to be modified.

To understand what needs to be modified, we need to return to Newton’s 2nd Law, . The only way for the winds to affect the vehicle’s motion, which is governed by its acceleration, is to somehow affect the net force on the vehicle. The forces acting on a vehicle are thrust, drag, lift, and weight. The only forces that the wind can affect are lift and drag, components of the total aerodynamic force.

How does the wind affect the aerodynamic forces? Imagine a vehicle flying in straight and level flight with no acceleration and no wind. Under these conditions weight is balanced by lift and drag is balanced by thrust. Now imagine a tail wind suddenly comes up. The velocity of the air with respect to the vehicle now decreases. Lift and drag depend on the relative air speed not the ground speed. If thrust does not change, there is now an unbalanced force and the vehicle accelerates forward until the relative speed is high enough for thrust and drag to balance again.

Although not the focus of this document, there is a whole study of how wind affects vehicles. Two specific areas are gust analysis and wind shear. Because aircraft have a finite speed, different parts enter gusts or wind shear at different times. This can not only affect angle-of-attack but also dynamic pressure.

Before we go any further, we have to understand reference frames. I like to use subscript with fractions notation. The reason I do this will become clear. With wind, there will be three objects to consider: the earth, wind, and the aircraft. Their identifications will be E, W, and A. The notation implies something of X with respect to Y. Now the following is true.

The subscripts with fractions notation allows me to write this down quickly without thinking. It works in the same way fractions multiply.

The velocity vector on the left gets the A/E subscript (velocity of the aircraft wrt earth), the first vector on the right side gets A/W (velocity of the aircraft wrt wind), and the last vector gets W/E (velocity of wind wrt earth). The velocity of the aircraft wrt wind is the air speed that the aerodynamic forces depend on. Then

This is a vector equation so the subtraction has to be done vectorally. To do this, I will define i to be the east direction, j to be in the north direction, and k to be the up direction. Note this is different than Vinh’s notation. The velocity of the vehicle wrt to the earth is given by

If the wind has the same positive directions as the vehicle but no vertical component then

Doing the vector subtraction and using Pythagorean theorem to get the magnitude gives

This is the speed that is used to calculate the aerodynamic forces in the equations of motion. Any place else where speed is needed the inertial speed is used.

Winds on a Real Vehicle

On a real vehicle, winds will affect the angle-of-attack and sideslip angle. To analyze these effects, equations are needed to describe the pitch, roll, and yaw moments about the center-of-gravity. To simplify the analysis, only pitching moment will be considered.

The equations are based on the net torque being equal to the product of the inertia matrix and the vector of angular accelerations about the vehicle’s axes. For a flat earth, the flight path angle and angle-of-attack uniquely specify the pitch angle in inertial coordinates. However, when considering the earth to not be flat, another angle is required. This is because the flight path angle is defined in a local tangential plane. To give an example, consider the space shuttle in LEO. As it orbits the earth, the top of the shuttle is always oriented towards the earth. The only way for this to happen is for the vehicle to have a constant angular velocity (pitch rate) that is equal to , approximately. This angular velocity is imparted during boost as the gimbal controllers drive the flight path angle to zero for a circular orbit.

Vinh calls this extra angle the range angle, and it is simply the angle that is swept out in a given time. Vinh’s full expression is

where

range angle

The l terms are the moment arms along the vehicle longitudinal axis. The M term is the aerodynamic pitching moment. The K term is the moment due to the mass flow of the gas ejected from the propulsion system. How to evaluate this is not known so it will be assumed to be zero. Be careful with signs and angle definitions. They may not be the same as you are used to.

Because we are not interested in the range angle or its derivatives except in the pitching moment equation, let’s separate this term. To do this we need to evaluate its second derivative wrt time. It first derivative wrt time is simply the angular velocity of the vehicle about earth.

By applying the quotient rule for the second derivative

Each term can be obtained from the existing equations of motion. If we combine all of the torque terms and rearrange the pitching moment equation

Now assume that you already have 9 EOMs programmed in. Define

Now just include the two right equations in the existing equations of motion. To get angle-of-attack for the next time step

If the angle-of-attack derivative is needed for a gimbal controller then

Estimating Normal and Axial Forces During Boost

Approximate the normal force by using the drag of a cylinder in cross flow. The speed of the cross flow is given by

Knowing the size of the booster, speed of the cross flow, and the altitude gives the Reynolds number which gives the normal force coefficient (drag coefficient of the cylinder). The normal force is then

The product of average diameter D and booster length L gives the proper reference area if the cylinder drag coefficient is referenced to diameter of the cylinder.

In the absence of other data, the aerodynamic pitching moment can be approximated by assuming that the cross flow produces a constant distributed normal force. For such a distribution, the pitching moment is given by

The CG along the longitudinal axis is measured from the bottom of the booster. Note that the CG changes as fuel is burned or stages are dropped. Be careful of sign conventions. I do not know if this agrees with Vinh.

The axial force coefficient needs to be calculated differently depending on speed. Subsonically, you can calculate the skin friction based on Reynolds number, based on booster length not diameter. You’ll need the wetted area which is . Supersonically you also need to include the wave drag. Hypersonically, use the Newtonian approximations.