Flight Mechanics Angles#

To describe the aircraft attitude a set of angles are used called Euler angles - these will be fully described in Module 3. The aircraft pitch is defined as the angle between the horizontal and the earth axis, and is given the symbol $\theta$.

To describe the aircraft flight path a set of angles describing the orientation of the aircraft velocity vector with respect to the earth are used. These are the flight path angle $\gamma$, and the track angle, $\tau$.

To describe the aerodynamic incidence, the aerodynamic angles $\alpha$ and $\beta$ are used. $\alpha$ is angle of attack, and $\beta$ is sideslip.

The angles above will be fully explored and utilised in the later modules, but an appreciation of the longitudinal angles is required here:

Aircraft Performance#

In aircraft performance, the climb or glide angle is denoted by $\theta$ in numerous texts - for example - and this angle represents the angle between the aircraft flight direction and the horizontal plane.

Looking at Fig. 20, you will see that this should really be denoted by $\gamma$. In the following, the glide or climb angle will be denoted by $\gamma$, but you should be aware that in most texts you will see it denoted by $\theta$.

In reality, $\gamma=\theta$ if and only if (iff) $\alpha=0$.

The distinction is made because this course later requires us to use $\theta$, and I don’t want you getting confused about what the angles mean.

Since the flight path angle, $\gamma$, is defined as positive in a glide and negative in a climb (positive $\gamma$ gives positive $W_e$ which is velocity towards the earth), it leads to requiring a negative angle continually in the climb expressions. uses a good approach and introduces

to avoid the negative expression. For the following, $\bar{\gamma}$ will be used.

Glide Angle#

In Fig. 21, the forces are parallel and perpendicular to the flight path as before, but you should note that this is no longer parallel with the ground. In the horizontal direction, equilibrium gives $W\cdot\sin\bar{\gamma}=D$ and in the vertical direction, equilibrium gives $L=W\cdot\cos\bar{\gamma}$. Hence the glide angle can be determined from simple trigonometry:

So it can be seen that the best/shallowest glide (the smallest $\bar{\gamma}$) occurs at the minimum drag to lift ratio - i.e., at $C_{L,md}$.

The glide angle dictates the glide distance, which is very important to be aware of in the cockpit in a glider as this dictates the furthest distance that can be covered along the ground.

The aircraft speed equation can be used with $L=W\cdot\cos\bar{\gamma}$ in place of $L=W$ - which shows that the speed that the shallowest glide occurs is not $V=V_{md}=\left[\frac{B}{A}\right]^\frac{1}{4}$ as $V_{md}$ as calculated previously is only valid for cruise.

import numpy as np
import plotly.graph_objects as go

# Make a vector of reasonable glide ratios - say 30 - 80
glide_ratios = np.linspace(2, 80, 1000)
glide_angles = np.arctan(1/glide_ratios)

Vbestglide_on_Vmd = np.sqrt(np.cos(glide_angles))


fig = go.Figure()
fig.add_trace(go.Scatter(x=glide_ratios, y=glide_angles*180/np.pi))

fig.update_layout(
    title="Glide Angle vs. Glide Ratio",
    xaxis_title="Glide Ratio",
    yaxis_title="Glide Angle / deg",
)

fig.show()

fig = go.Figure()
fig.add_trace(go.Scatter(x=glide_ratios, y=Vbestglide_on_Vmd))

fig.update_layout(
    title="$\\text{Ratio of best glide speed and }V_{md}$",
    xaxis_title="Glide Ratio",
    yaxis_title="$\\frac{V_{bestglide}}{V_{md}}$",
)

fig.show()

Sink rate#

Whilst the glide angle dictates the distance along the ground (hence giving the range), the best endurance is given by the slowest overall descent. Thus the sink rate needs to be minimised.

The sink rate, $V_{sink}$, is the vertical component of flight speed - $V_{sink}=V\cdot\sin\bar{\gamma}$. From the reasoning used previous regarding the exchange of potential energy for kinetic energy, you should be able to intuit that the minimum sink rate occurs at the minimum power speed.

The sink rate is the vertical component of flightspeed, $V_{sink}=V\cdot\sin\bar{\gamma}$ - see Fig. 22

so

from equilibrium and making the small angle assumption for $\bar{\gamma}$

using the aircraft speed equation and using the small angle assumption to substitute $L\simeq W$

since the other terms are just constants, or outside of our control, then the minimum sink rate is found at the minimum of $\frac{C_D}{C_L^{3/2}}$ which can be found from some elementary calculus by substituting $C_D=C_{D0}+K\cdot C_L^2$ and differentiating by $C_L$:

Substitution of this into the aircraft speed equation, Eq. (5), yields the speed for minimum sink

which is equal to $V_{minsink}=\left[\frac{B}{3\,A}\right]^\frac{1}{4}$, or the minimum power condition, which is what was expected from intuition.

Glider Polar#

For the pilot, as has been mentioned previously, the actual lift coefficient or angle of attack for a given condition is not usually of relevance. A glider pilot will be keenly aware of their best sink rates, and will desire to know the airspeed(s) at which these occur.

Glider manufacturers tend to provide a chart of sink rate vs. airspeed, which is proportional to the $\frac{D}{L}$ ratio against the inverse of the lift coefficient (with some additional exponents thrown in for good measure).

# Aircraft Input data - representative glider data taken from https://booksite.elsevier.com/9780123973085/content/APP-C4-DESIGN_OF_SAILPLANES.pdf page 23
# and from Stemme S10 info at https://en.wikipedia.org/wiki/Stemme_S10
#
# Note that I have reduced the stall speed slightly to show the shape of the drag polar in more detail
CD0 = 0.008
e = 0.95
AR = 28
W = 850 * 9.80665 # Gross weight
Vstall = 35 # knots
Vne = 146 # knots
S = 18.7
rho_sl = 1.2255

# Get the induced drag factor from Oswold efficiency
K = 1/np.pi/e/AR


# Make a vector of speeds
Vknots = np.linspace(Vstall, Vne, 1000) # In knots
Vms = Vknots * 0.5144444 # In m/s

# Get the CL values for these speeds
CLs = 2*W/rho_sl/S/Vms**2

# Get the corresponding drags
CDs = CD0 + K * CLs**2



# Determine the sink rates
w_e = np.sqrt(2 * W / rho_sl / S) * CDs / CLs**(3/2) / 0.5144444

#### Get points for min glide angle
# Get ClMD and associated sink rate for plotting
A = .5 * CD0 * rho_sl * S
B = K * W**2 / rho_sl / 0.5 / S

# Minimum drag speed
Vmd_ms = (B/A)**(1/4)
Vmd_knots = (B/A)**(1/4) / 0.5144444

# Associated Cl and Cd
Clmd = np.sqrt(CD0/K)
Cdmd = CD0 + K * Clmd**2

# Sink rate at minimum drag speed
w_emd = np.sqrt(2 * W / rho_sl / S) * Cdmd / Clmd**(3/2) / 0.5144444

#### Do the same for the minimum sink rate
# Minimum power speed
Vmp_ms = (B/3/A)**(1/4)
Vmp_knots = (B/3/A)**(1/4) / 0.5144444

# Associated Cl and Cd
Clmp = np.sqrt(3*CD0/K)
Cdmp = CD0 + K * Clmp**2

# Sink rate at minimum power speed
w_emp = np.sqrt(2 * W / rho_sl / S) * Cdmp / Clmp**(3/2) / 0.5144444

# Make a plot
fig = go.Figure()
fig.add_trace(go.Scatter(x=Vknots, y=w_e, name=f"Gross weight = {W}N"))



# Annotate it
fig.add_trace(go.Scatter(x=[Vmd_knots], y=[w_emd], mode="markers+text", text=f"Vmd={Vmd_knots:1.2f}kn$", textposition="middle right", name="Annotation"))
fig.add_trace(go.Scatter(x=[Vmp_knots], y=[w_emp], mode="markers+text", text=f"Vmp={Vmp_knots:1.2f}kn$", textposition="middle right", name="Annotation"))

# Draw a line to the minimum drag speed
fig.add_trace(go.Scatter(x=[0, Vmd_knots*2], y=[0, w_emd*2], mode='lines', name="Annotation"))


fig.update_layout(
    title="Glider Polar",
    xaxis_title="Airspeed kn EAS",
    yaxis_title="Sink rate kn EAS"
)


fig.update_yaxes(range=[2.5, 0])
fig.update_xaxes(range=[0, 150])

for trace in fig['data']: 
    if(trace['name'] == "Annotation"): trace['showlegend'] = False

fig.show()

You can see that an increase in the aircraft weight moves the curve along so its tangent remains the same.

Note: The weights for the plot are likely unfeasible for this aircraft but they make for a nice plot

Powered Descent#

The procedure for powered descent and determination of descent slope and sink rate is the same but with $D=T + W\sin\gamma$