Angle of climb at Vx and glide angle when "overweight": five questions

I did more brainstorming at work before going home early with a
headache...this is regarding the last point in my post.

Just to reiterate, I wanted to know why your best glide angle was
independent of weight whereas your maximum angle of climb was.

Here's what I came up with with a FBD:

For an aircraft climbing, unaccelerated, where theta is the flight
path angle.

T = D + W*sin(theta)
L = W*cos(theta)

For small theta, cos(theta) = 1 and sin(theta) = theta

so we now have

T = D + W*theta
L = W

flight path angle is given as (T - D)/W [Eq. 1]

or T/W - D/W or T/W - D/L or

theta = T/W - 1/(CL/CD) [Eq. 2]

First, you'll notice that the first term of Eq. 2 is the thrust to
weight ratio and the second is the lift to drag ratio. To increase
theta during a climb or minimize theta during a glide, one must
minimize the second term by flying at best L/D AOA. For a given
thrust, decreasing weight will increase theta.

The equation above would apply for both climbs and descents. What's
interesting to note is that in a glide when T = 0, Eq. 2 simplifies to
theta = - 1/(CL/CD), independent of airspeed or weight. The
dependence on load factor mathematically vanishes when the engine goes
out. How strange!

However, when flying with engine thrust and at the best (L/D) AOA, an
increase in load factor will IMPACT theta, lessening the climb angle
or making it more "negative".

Don't you find it coincidental that the absence of thrust makes the
dependence of load factor upon theta vanish? Can it only be explained
from the math?

So somehow, the magic equations seem to explain my confusion...but I
am not quite at peace. Au contraire. I've only talked about L/D,
weight, and thrust. What about drag? I am still claiming that you'll
experience an increase in drag with a weight increase REGARDLESS of
whether or not you have thrust. Thus, due to the increase in drag and
REGARDLESS of thrust or not, your angle THETA will be affected (i.e.
theta will get larger, more negative, airplane will pitch DOWN). In
the case of gliding flight, everybody but me agrees that your glide
angle won't be affected. Obviously, I am wrong but I don't know why.
To reiterate, that's why I've been writing all this stuff, 'cause I
don't get why THETA happens to change only when you have thrust but
doesn't when you don't have any and are gliding.

Alright, so the increase in drag with respect to load factor is shown
as:

D = V^2*CD where CD = (1 + CL^2)

By virtue of increasing your load factor (either via CL and/or V),
your drag goes up...[right?]

Now, try to tie that in to Eq. 2 in a desperate attempt to explain why
an increase in DRAG won't cause a increased glide angle (less range)
but will cause a logical decrease in climb angle.

From D = V^2 * (1 + CL^2) and theta = T/W - 1/(CL/CD)

Well, I'll stop here because I seemed to have taken the wrong path, as
I am pretty much stuck. ARRGHH! I AM GOING TO KILL A TURKEY!

I didn't go through all your math, but consider this:

Your engine makes the same power regardless of weight, so the higher
the weight the slower the climb.

However, as you get heavier you have stored more energy by climbing
(that's one reason why it took you longer with the same engine power).
With more weight, you have more stored energy per 100 feet. At a
higher airspeed you are also descending faster.

As you lower a weight you are releasing energy, the rate at which this
is released is effectively power (this is what maintains your airspeed
without the engine.) So your heavier weight and higher rate of descent
gives you more "power" in your glide.

Read a book. I think you'll find the two main factors in climb are power
to weight ratio and span loading. Guess what, more weight hurts both.

In article ,
(Koopas Ly) wrote:
I did more brainstorming at work before going home early with a
headache...this is regarding the last point in my post.

Just to reiterate, I wanted to know why your best glide angle was
independent of weight whereas your maximum angle of climb was.

Here's what I came up with with a FBD:

For an aircraft climbing, unaccelerated, where theta is the flight
path angle.

T = D + W*sin(theta)
L = W*cos(theta)

For small theta, cos(theta) = 1 and sin(theta) = theta

so we now have

T = D + W*theta
L = W

flight path angle is given as (T - D)/W [Eq. 1]

or T/W - D/W or T/W - D/L or

theta = T/W - 1/(CL/CD) [Eq. 2]

First, you'll notice that the first term of Eq. 2 is the thrust to
weight ratio and the second is the lift to drag ratio. To increase
theta during a climb or minimize theta during a glide, one must
minimize the second term by flying at best L/D AOA. For a given
thrust, decreasing weight will increase theta.

The equation above would apply for both climbs and descents. What's
interesting to note is that in a glide when T = 0, Eq. 2 simplifies to
theta = - 1/(CL/CD), independent of airspeed or weight. The
dependence on load factor mathematically vanishes when the engine goes
out. How strange!

However, when flying with engine thrust and at the best (L/D) AOA, an
increase in load factor will IMPACT theta, lessening the climb angle
or making it more "negative".

Don't you find it coincidental that the absence of thrust makes the
dependence of load factor upon theta vanish? Can it only be explained
from the math?

So somehow, the magic equations seem to explain my confusion...but I
am not quite at peace. Au contraire. I've only talked about L/D,
weight, and thrust. What about drag? I am still claiming that you'll
experience an increase in drag with a weight increase REGARDLESS of
whether or not you have thrust. Thus, due to the increase in drag and
REGARDLESS of thrust or not, your angle THETA will be affected (i.e.
theta will get larger, more negative, airplane will pitch DOWN). In
the case of gliding flight, everybody but me agrees that your glide
angle won't be affected. Obviously, I am wrong but I don't know why.
To reiterate, that's why I've been writing all this stuff, 'cause I
don't get why THETA happens to change only when you have thrust but
doesn't when you don't have any and are gliding.

Alright, so the increase in drag with respect to load factor is shown
as:

D = V^2*CD where CD = (1 + CL^2)

By virtue of increasing your load factor (either via CL and/or V),
your drag goes up...[right?]

Now, try to tie that in to Eq. 2 in a desperate attempt to explain why
an increase in DRAG won't cause a increased glide angle (less range)
but will cause a logical decrease in climb angle.

From D = V^2 * (1 + CL^2) and theta = T/W - 1/(CL/CD)

Well, I'll stop here because I seemed to have taken the wrong path, as
I am pretty much stuck. ARRGHH! I AM GOING TO KILL A TURKEY!

On Thu, 27 Nov 2003 at 15:34:14 in message
.net, cddb
wrote:

So somehow, the magic equations seem to explain my confusion...but I
am not quite at peace. Au contraire. I've only talked about L/D,
weight, and thrust. What about drag? I am still claiming that you'll
experience an increase in drag with a weight increase REGARDLESS of
whether or not you have thrust. Thus, due to the increase in drag and
REGARDLESS of thrust or not, your angle THETA will be affected (i.e.
theta will get larger, more negative, airplane will pitch DOWN). In
the case of gliding flight, everybody but me agrees that your glide
angle won't be affected. Obviously, I am wrong but I don't know why.
To reiterate, that's why I've been writing all this stuff, 'cause I
don't get why THETA happens to change only when you have thrust but
doesn't when you don't have any and are gliding.

By rearranging the simple equations I may be able to help you with that
one.

W= weight, L = Lift, D = Drag, t = Thrust, theta = angle of climb (if
negative implies descent, V = forward speed

You started with L = W*cos (theta)
and T
= D + W*sin(theta)

For a change eliminate W from the above
by taking W = L/(cos(theta)

Then T = D + L*tan(theta)

Let K = 0.5*density* wing area

Then T = K*V^2*Cd + K*V^2*Cl Divide both sides by K*V^2

T/(K*V^2) = (Cd + Cl*tan(theta))..........................[1]

Now if T= 0 the left hand side must be 0 and it follows that

Cd = - Cl*tan(theta) this means that theta must be negative and

tan(theta) = -Cd/Cl as one would expect.

The reverse of this is that positive or negative thrust is always needed
_except_ when Tan(theta) = -Cd/Cl

At any other climb angle, thrust or extra drag would be needed for
balance. To glide more steeply at the same Cl and Cd negative thrust
would be needed

Perhaps it does not help after all?

Also in [1] if theta = 0 then

T = K*V^2*Cd which is a necessary condition for steady level flight..




--
David CL Francis