Determination of bank angle from speed and heading change per time period

[Grumman-581[_1_]]

Anyone know of a formula for determing the current bank angle, given the
current velocity and rate of change in heading over a particular period of
time? I was thinking of adding an artificial horizon to program that I'm
working on and figured that by taking the current velocity as reported by
the GPS and the change in heading as reported by the last two
course-over-ground measurements from the GPS, I should be able to
determine a close approximation of the actual bank angle... Of course,
this assumes that the current winds are ignored in addition to assuming a
coordinated turn... The current system that I am using only gets updates
at 1 Hz which might be a bit low, but I'm thinking of getting the Garmin
18 5 Hz unit which should be sufficiently fast in its update rate...

[d&tm]

"Grumman-581" > wrote in message
...
> Anyone know of a formula for determing the current bank angle, given the
> current velocity and rate of change in heading over a particular period of
> time? I was thinking of adding an artificial horizon to program that I'm
> working on and figured that by taking the current velocity as reported by
> the GPS and the change in heading as reported by the last two
> course-over-ground measurements from the GPS, I should be able to
> determine a close approximation of the actual bank angle... Of course,
> this assumes that the current winds are ignored in addition to assuming a
> coordinated turn... The current system that I am using only gets updates
> at 1 Hz which might be a bit low, but I'm thinking of getting the Garmin
> 18 5 Hz unit which should be sufficiently fast in its update rate...


bank angle = tan ^-1 ( 2*PI*v / g *t )

v= velocity
g= gravitational acceleration
t = time for 360 degree turn rate
PI =3.14159
make sure you use consistant units for v, g and t
for SI use v in m/s
g = 9.8 m/s^2
t in seconds

in imperial g = 32 ft /s^2 ( If I remember correctly)
v in ft/sec
t in sec

Terry
PPL downunder

[Bob Moore]

Grumman-581 wrote
> Anyone know of a formula for determing the current bank angle, given
> the current velocity and rate of change in heading over a particular
> period of time?

I think that you are going to have to work with TAS instead of GS,
but Figure 2.29 on page 179 of Aerodynamics for Naval Aviators has
a graph of all the factors involved.

Bob Moore

[Grumman-581[_1_]]

On Tue, 09 Jan 2007 13:17:32 +0000, in
8>, Bob Moore wrote:
> I think that you are going to have to work with TAS instead of GS, but
> Figure 2.29 on page 179 of Aerodynamics for Naval Aviators has a graph
> of all the factors involved.

Yeah, to be perfectly accurate, I would need air speed and not ground
speed in addition to wind angle corrections, but I was curious if I could
get a good approximation of the bank angle from just what I'm getting from
the GPS without having to add a considerably costlier device like an
accelerometer / gyro with rs232 outputs... The Garmin 18 5Hz puck type GPS
goes for around $150... A bit more than the 1Hz units that are typical for
car navigation, but not entirely out of line for this project... The
accelerometer / gyro systems that I've seen so far with rs232 outputs go
for around $1K-$2K... I understand that there is some new work in
accelerometers for PC-based pointing devices that have brought the prices
down considerably compared to the previous devices like this:

http://www.watson-gyro.com/products/rate_gyro_ARS-E322-1A_80_spec.html

[Grumman-581[_1_]]

On Wed, 10 Jan 2007 00:03:22 +1100, in >,
d&tm wrote:
> bank angle = tan ^-1 ( 2*PI*v / g *t )
>
> v= velocity
> g= gravitational acceleration
> t = time for 360 degree turn rate
> PI =3.14159
> make sure you use consistant units for v, g and t
> for SI use v in m/s
> g = 9.8 m/s^2
> t in seconds
>
> in imperial g = 32 ft /s^2 ( If I remember correctly)
> v in ft/sec
> t in sec

Thanks Terry... Couple of questions though...

1. Is the resulting bank angle in radians?
2. Does the formula for the bank angle actually have "2*PI*v / g*t"
in it or was it supposed to be "2*PI*v / (g*t)"?

[[email protected]]

Grumman-581 > wrote:
> The accelerometer / gyro systems that I've seen so far with rs232
> outputs go for around $1K-$2K...

You can get a 3-axis accelerometer with analog output for around $300.
I recently used one like this
http://www.xbow.com/Products/productdetails.aspx?sid=185 for a project
in a ground vehicle. I used a three-axis one that takes power and gives
you three analog outputs X, Y, Z. The same company also sells an RS232
A/D board but IMHO it's a little spendy at $300. If USB is OK I know
you can get small USB DAQ boxes from several different vendors. I was
using an older National Instruments full length ISA card, but then again
it was OK for me to have an entire desktop PC case in the back seat of
the car. :)

None of this stuff is official aviation stuff (nice aluminum box, round
connectors, etc) but then again it doesn't have an official aviation
price tag either.

Standard disclaimers apply; I don't get money from any of the companies
mentioned.

Matt Roberds

[d&tm]

"Grumman-581" > wrote in message
...
> On Wed, 10 Jan 2007 00:03:22 +1100, in >,
> d&tm wrote:
> > bank angle = tan ^-1 ( 2*PI*v / g *t )
> >
> > v= velocity
> > g= gravitational acceleration
> > t = time for 360 degree turn rate
> > PI =3.14159
> > make sure you use consistant units for v, g and t
> > for SI use v in m/s
> > g = 9.8 m/s^2
> > t in seconds
> >
> > in imperial g = 32 ft /s^2 ( If I remember correctly)
> > v in ft/sec
> > t in sec
>
> Thanks Terry... Couple of questions though...
>
> 1. Is the resulting bank angle in radians?
> 2. Does the formula for the bank angle actually have "2*PI*v / g*t"
> in it or was it supposed to be "2*PI*v / (g*t)"?

The bank angle is in what ever units you want. it is the angle expressed as
degrees or radians that gives a tangent of 2*PI*v /( g*t)
(sorry I should have put in the brackets)
check your equation with a standard rate 1 turn. At 100kts a rate 1 turn
will have t =120 seconds and give a bank angle of 15.4 degrees.
terry

[Grumman-581[_1_]]

On Wed, 10 Jan 2007 18:31:29 +1100, in >,
d&tm wrote:
> The bank angle is in what ever units you want. it is the angle
> expressed as degrees or radians that gives a tangent of 2*PI*v /( g*t)
> (sorry I should have put in the brackets) check your equation with a
> standard rate 1 turn. At 100kts a rate 1 turn will have t =120 seconds
> and give a bank angle of 15.4 degrees.

I tried it both ways today and pretty much figured that it needed the
parentheses around the "g*t"... The atan function was returning things in
radians, so that's what gave me the brain fart...

I'm getting approximately 15.175 degrees for a bank angle, but that's
probably because I'm using a more accurate measurement for pi and g... Not
that it is likely to really matter since when I graph it, putting it to
the nearest degree will probably be accurate enough... Haven't decided if
I want to redraw the gauge each time or have a set of predrawn images that
I shift around indexed by the bank angle...

Thanks for the help...

--
Mike Shelley
N581 -- AA5A -- AXH

[d&tm]

"Grumman-581" > wrote in message
...
> On Wed, 10 Jan 2007 18:31:29 +1100, in >,
> d&tm wrote:
> > The bank angle is in what ever units you want. it is the angle
> > expressed as degrees or radians that gives a tangent of 2*PI*v /( g*t)
> > (sorry I should have put in the brackets) check your equation with a
> > standard rate 1 turn. At 100kts a rate 1 turn will have t =120 seconds
> > and give a bank angle of 15.4 degrees.
>
> I tried it both ways today and pretty much figured that it needed the
> parentheses around the "g*t"... The atan function was returning things in
> radians, so that's what gave me the brain fart...
>
> I'm getting approximately 15.175 degrees for a bank angle, but that's
> probably because I'm using a more accurate measurement for pi and g... Not
> that it is likely to really matter since when I graph it, putting it to
> the nearest degree will probably be accurate enough... Haven't decided if
> I want to redraw the gauge each time or have a set of predrawn images that
> I shift around indexed by the bank angle...
>
Its close but is it close enough at greater angles.? what are you using for
the conversion of speed.? I used g =9.8 Pi as 3.14159 and v = kts
*1852/3600 m/s
terry

[Chris]

"d&tm" > wrote in message
...
>
> "Grumman-581" > wrote in message
> ...
>> On Wed, 10 Jan 2007 00:03:22 +1100, in >,
>> d&tm wrote:
>> > bank angle = tan ^-1 ( 2*PI*v / g *t )
>> >
>> > v= velocity
>> > g= gravitational acceleration
>> > t = time for 360 degree turn rate
>> > PI =3.14159
>> > make sure you use consistant units for v, g and t
>> > for SI use v in m/s
>> > g = 9.8 m/s^2
>> > t in seconds
>> >
>> > in imperial g = 32 ft /s^2 ( If I remember correctly)
>> > v in ft/sec
>> > t in sec
>>
>> Thanks Terry... Couple of questions though...
>>
>> 1. Is the resulting bank angle in radians?
>> 2. Does the formula for the bank angle actually have "2*PI*v / g*t"
>> in it or was it supposed to be "2*PI*v / (g*t)"?
>
> The bank angle is in what ever units you want. it is the angle expressed
> as
> degrees or radians that gives a tangent of 2*PI*v /( g*t)
> (sorry I should have put in the brackets)
> check your equation with a standard rate 1 turn. At 100kts a rate 1 turn
> will have t =120 seconds and give a bank angle of 15.4 degrees.
> terry
As an approximation
speed in metres/second is half speed in kts eg 100kts is about 50m/s
actually it is 52m/s.

[Blueskies]

"Grumman-581" > wrote in message
...
: On Tue, 09 Jan 2007 13:17:32 +0000, in
: 8>, Bob Moore wrote:
: > I think that you are going to have to work with TAS instead of GS, but
: > Figure 2.29 on page 179 of Aerodynamics for Naval Aviators has a graph
: > of all the factors involved.
:
: Yeah, to be perfectly accurate, I would need air speed and not ground
: speed in addition to wind angle corrections, but I was curious if I could
: get a good approximation of the bank angle from just what I'm getting from
: the GPS without having to add a considerably costlier device like an
: accelerometer / gyro with rs232 outputs... The Garmin 18 5Hz puck type GPS
: goes for around $150... A bit more than the 1Hz units that are typical for
: car navigation, but not entirely out of line for this project... The
: accelerometer / gyro systems that I've seen so far with rs232 outputs go
: for around $1K-$2K... I understand that there is some new work in
: accelerometers for PC-based pointing devices that have brought the prices
: down considerably compared to the previous devices like this:
:
: http://www.watson-gyro.com/products/rate_gyro_ARS-E322-1A_80_spec.html
:

Don't know why wind angle would be important. Have to assume a coordinated turn also - a rudder turn would yaw like
crazy but not necessarily result in a bank angle....

[Grumman-581[_1_]]

On Thu, 11 Jan 2007 00:05:44 +1100, in >,
d&tm wrote:
> Its close but is it close enough at greater angles.? what are you using
> for the conversion of speed.? I used g =9.8 Pi as 3.14159 and v =
> kts *1852/3600 m/s

For conversion of speed, I'm using 6076 ft per nm, 3.14159265358979 for
pi, 9.80665 m/s^2 or 32.17398421 ft/sec^2 for g...

Interestingly on g, I'ver seen sources that quite it as being defined as
9.80665 *exactly* and others that quote it at 32.1740486... Not sure about
the later figure though since it is not exactly correct at least from a
conversion standpoint of the 9.80665 figure using the value of 39.37
inches per meter that I remembered... A quick lookup on the net and I find
that my memory was not correct and the 39.37 was not an *exact* figure...
Plugging in 39.37007874015748 instead and the values agree... For some
reason, I had thought that the 39.37 had been defined as an *exact*
value... Of course this also means that my memory is faulty on the 2.54
cm/in value also...

With regards to higher velocities and decreased values of time for the 360
degree circle, even with a 1 second 360 degree circle and 500 kts, the
values match to more digits than I really need (i.e. 88.26219913 and
88.26219912 degrees respectively for SI and US measurement units)... Yeah,
I would like to have an exact match, but there's probably some conversion
factor that I'm not using enough significant digits with...

[Danny Deger]

"Grumman-581" > wrote in message
...
> On Thu, 11 Jan 2007 00:05:44 +1100, in >,
> d&tm wrote:
snip

> Plugging in 39.37007874015748 instead and the values agree... For some
> reason, I had thought that the 39.37 had been defined as an *exact*
> value... Of course this also means that my memory is faulty on the 2.54
> cm/in value also...

My understanding is an inch is EXACTLY 2.54 cm.

Danny Deger

[[email protected]]

Grumman-581 > wrote:
> For conversion of speed, I'm using 6076 ft per nm, 3.14159265358979 for
> pi, 9.80665 m/s^2 or 32.17398421 ft/sec^2 for g...

Whoever is selling you your floats, doubles, and long doubles must be
making a killing.

> Interestingly on g, I'ver seen sources that quite it as being defined
> as 9.80665 *exactly* and others that quote it at 32.1740486...

http://en.wikipedia.org/wiki/Gravity_%28Earth%29 has the definition of
9.80665 m/(s^2). It does vary with where you are on the planet as well
as how far you are above it.
http://en.wikipedia.org/wiki/Acceleration_due_to_gravity has a few more
details.

> For some reason, I had thought that the 39.37 had been defined as an
> *exact* value...

http://en.wikipedia.org/wiki/Inch#International_inch says that the yard
is defined as 0.9144 m. 0.9144 is exactly 36 * 2.54, so the correct
statement may be that an inch is exactly 2.54 cm. 36/0.9144 is
39.370079 according to the calculator; rounding off to 39.37 is a
whopping 0.0002% error.

Standard disclaimers about using Wikipedia as a reference apply.

> With regards to higher velocities and decreased values of time for the
> 360 degree circle, even with a 1 second 360 degree circle and 500 kts,
> the values match to more digits than I really need (i.e. 88.26219913
> and 88.26219912 degrees respectively for SI and US measurement units)...

Your protractor salesman must also be living quite well. :)

Matt Roberds

[Grumman-581[_1_]]

On Thu, 11 Jan 2007 14:33:45 -0600, in
>, Danny Deger wrote:
> My understanding is an inch is EXACTLY 2.54 cm.

Yeah, well I thought that a meter was defined as EXACTLY 39.37 inches
also, so I was leaving myself an out, just in case yet another previoiusly
held belief turned out to be wrong... According to
http://www.pmel.org/Handbook/HBConversion.htm, it appears that 2.54 is an
exact figure... Damn, that's a surprise...

[Grumman-581[_1_]]

On Thu, 11 Jan 2007 22:43:13 +0000, in >,
mroberds wrote:
> Whoever is selling you your floats, doubles, and long doubles must be
> making a killing.

I get a discount on them at TWORD-Depot... <grin>

Better to have the accuracy and choose to not use it than to need it and
not have it... If performance becomes an issue, you can always decrease
the bytes allocated for your variables...

[Don Byrer]

On Tue, 09 Jan 2007 02:58:02 -0600, Grumman-581
> wrote:

>Anyone know of a formula for determing the current bank angle, given the
>current velocity and rate of change in heading over a particular period of
>time? I was thinking of adding an artificial horizon to program that I'm
>working on and figured that by taking the current velocity as reported by
>the GPS and the change in heading as reported by the last two
>course-over-ground measurements from the GPS, I should be able to
>determine a close approximation of the actual bank angle... Of course,
>this assumes that the current winds are ignored in addition to assuming a
>coordinated turn... The current system that I am using only gets updates
>at 1 Hz which might be a bit low, but I'm thinking of getting the Garmin
>18 5 Hz unit which should be sufficiently fast in its update rate...

Sounds like a great project...

Provide a great deal of valuable "darn-near-real-time" info to a
pilot...and at a cost that wont break the bank.

Makes sense...therefore...
(do I even NEED to finish the comment?)

MORE POWER TO YA!

Let us know how it works out.

--Don Byrer

Don Byrer KJ5KB
Radar Tech & Smilin' Commercial Pilot Guy
Glider & CFI wannabe
kj5kb-at-hotmail.com

"I have slipped the surly bonds of earth; now if I can just land without bending the gear..."
"Watch out for those doves...<smack-smack-smack-smack...>"