Total Angle of Attack

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

The total angle of attack $\eta$ is the angle between the velocity axis $o_1x_v$ and the vehicle's longitudinal axis $o_1x_1$ . In the total angle-of-attack plane (the $x_1o_1x_v$ plane), the aerodynamic force $\boldsymbol{R}$ can be decomposed along the longitudinal axis into the axial force $X_1$ and total normal force $N$ , or along the velocity axis into the drag $X$ and total lift $L$ .

Core Elements

Relationship with Angle of Attack and Sideslip Angle

The exact relationship between the total angle of attack and the angle of attack $\alpha$ and sideslip angle $\beta$ :

\cos\eta = \cos\alpha \cdot \cos\beta

or equivalently:

\sin^2\eta = \sin^2\alpha + \sin^2\beta - \sin^2\alpha \cdot \sin^2\beta

When $\alpha$ and $\beta$ are small angles, the approximate relationship is:

\eta = \sqrt{\alpha^2 + \beta^2}

Force Decomposition Relationships

In the total angle-of-attack plane, the relationships between the axial force $X_1$ , total normal force $N$ and the drag $X$ , total lift $L$ :

\left\{\begin{array}{l} X = N\sin\eta + X_1\cos\eta \\ L = N\cos\eta - X_1\sin\eta \end{array}\right.

The corresponding coefficient relationships:

\left\{\begin{array}{l} C_x = C_N\sin\eta + C_{x1}\cos\eta \\ C_L = C_N\cos\eta - C_{x1}\sin\eta \end{array}\right.

Relationship Between Total Normal Force and Component Forces

The relationship between the total normal force $N$ and the normal force $Y_1$ , lateral force $Z_1$ :

N = \sqrt{Y_1^2 + Z_1^2}

The relationship between the total lift $L$ and the lift $Y$ , side force $Z$ :

L = \sqrt{Y^2 + Z^2}

Physical Significance

The total angle of attack comprehensively represents the spatial orientation relationship between the vehicle's velocity vector and body axis. When the sideslip angle $\beta = 0$ , the total angle of attack equals the absolute value of the angle of attack. The concept of total angle of attack simplifies the expression of aerodynamic forces, enabling the reentry equations of motion to be expressed uniformly using the total angle of attack and total lift, applicable to all types of reentry vehicles.

Application Value

The total angle of attack is the core parameter for reentry vehicle aerodynamic analysis. It enables the transformation of a three-dimensional aerodynamic problem into a two-dimensional problem within the total angle-of-attack plane, simplifying the derivation of reentry equations of motion. The total angle of attack is a key variable in trim angle-of-attack design, reentry corridor determination, and lift control.

References

Zheng W, An X Y, Zhou X, He R Z. Aerospace Flight Mechanics[M]. National University of Defense Technology, 2026.
Jia P R, Chen K J, et al. Long-Range Rocket Ballistics[M]. National University of Defense Technology Press.