Rotary Wing Forum speed derivative Xu
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#1
01-06-2017, 10:09 AM
 kolibri282 Platinum Member Join Date: Aug 2009 Location: Duesseldorf Posts: 2,458
speed derivative Xu

In the Glasgow report "Application of Parameter Estimation to Improved Autogyro Simulation Model Fidelity" table 1 shows the speed derivative Xu (here it is called drag derivative, see fig. 9a) contributions from various components. It was this table that prompted my interest in propeller analysis programs (see: http://www.rotaryforum.com/forum/showthread.php?t=46870 )
I have currently no model for the Montgomerie Parsons propeller, so can not really say whether an analysis would reproduce those values but for the Bensen propeller from post #1 things are quite different. While the speed derivative becomes larger with speed for the MP prop it gets smaller for the Bensen (see attachment #1). The tangent to the curve has a negative slope at every point. Image #2 shows a graph of finite differences I calculated from curve 1. I think this is due to the fact that the pitch of the Bensen propeller was selected such that the aircraft would not exceed a speed of about 60 mph. If the last finite difference value (V=55mph) of about -5 is normalized by aircraft mass is gives a value of -0.02 which is close to what my program calculates. For lower speeds my values increase (-0.04 at 25mph) but are much smaller than what the propeller thrust derivatives suggest (-0.144 at 25mph)
Attached Thumbnails

__________________
Cheers,

Juergen

..Il semble que la perfection soit atteinte..
....non quand il n'y a plus rien à ajouter,...
...mais quand il n'y a plus rien à retrancher...
- Antoine de Saint-Exupéry -

Last edited by kolibri282; 01-06-2017 at 10:50 AM.
#2
01-07-2017, 01:28 AM
 Jean Claude Platinum Member Join Date: Jan 2009 Location: Centre FRANCE Posts: 1,568

Juergen,
The slope of the thrust curve on this graph assumes the continuous throttle change (*), which is not the assumption used to limit the forward speed to 60 mph.
With constant throttle, the thrust slope of propeller is smaller.

(*)The proof is provided by the great decreasing of power as the increasing forward speed, while it lightly increases if constant throttle. Here is just an example:

Last edited by Jean Claude; 01-07-2017 at 05:21 AM.
#3
01-07-2017, 06:18 AM
 Dogcowdaddy Member Join Date: Apr 2011 Location: St. Leonard, Md, & Tampa, FL Posts: 13

The derivative Xu is the product of a math model linearized by assuming small perturbations about a trim condition. Its definition is:

Change in X force per unit change in velocity/mass, evaluated at the trim condition.

As such the evaluation of force change with velocity must be at thrust for level flight (TLF) at the trim condition (throttle fixed). In the fixed wing world Xu is the main damping derivative for the Phugoid mode.

FWIW

Jim
#4
01-07-2017, 09:51 PM
 kolibri282 Platinum Member Join Date: Aug 2009 Location: Duesseldorf Posts: 2,458

Thank you for taking part in this thread Jean-Claude and your input! As a last effort I have last night used an analytical expression from Roscam to calculate the propeller derivatives. The values I got were even a bit lower than the ones my program calculates but for half the speed I got twice the value which is pretty much what my program predicts. I have run out of ideas and will leave it at that now. I have no idea how the Glasgow team has arrived at the values they present in their report.
__________________
Cheers,

Juergen

..Il semble que la perfection soit atteinte..
....non quand il n'y a plus rien à ajouter,...
...mais quand il n'y a plus rien à retrancher...
- Antoine de Saint-Exupéry -

Last edited by kolibri282; 01-08-2017 at 02:36 AM.
#5
01-08-2017, 05:34 AM
 Jean Claude Platinum Member Join Date: Jan 2009 Location: Centre FRANCE Posts: 1,568

Juergen,
With the characteristics of Montgomerie Parson's gyroplane G-UNIV (355 kg) , I obtains at 25 mètres per second (25 m/s, 56mph):
Xu due to propeller: +0.06 N/m/s by kg (the thrust decreases as the forward speed increases)
Xu due to aiframe : +0.05 N/m/s by kg (the airframe drag increases as the forward speed increases)
Xu due to rotor : - 0.09 N/m/s by kg (the rotor drag decreases as the forward speed increases)

Table 1 (post #1) makes me suppose a propeller too weakly pitched.
Three blades too wide require a small pitch to reach the correct rpm of the engine. And the high static thrust giving the illusion of a better thrust in flight.

Last edited by Jean Claude; 01-08-2017 at 05:51 AM.
#6
01-08-2017, 05:48 AM
 kolibri282 Platinum Member Join Date: Aug 2009 Location: Duesseldorf Posts: 2,458

My values are a bit smaller than yours, Jean-Claude, but your Xu prop value is still only about half the value from the report, so we both seem to conclude that the report values are doubtful. Could you please explain a bit more how you obtained the 0.06 for the Xu prop value? Thank you!
#7
01-08-2017, 07:27 AM
 Jean Claude Platinum Member Join Date: Jan 2009 Location: Centre FRANCE Posts: 1,568

Too many values in the report are doubtful to grant a credibility to it author, Juergen. Rotor Xu of same sign that airframe is a proof he not credible, because the drag of the rotor decreases as the forward speed increases, while the airframe drag increases.

On the Xu prop.
Here is the thrust gived by JavaProp for three blades of diameter 1.57 m at 2500 rpm with 65 hp, and we can suppose the green curve to 2/3 power to level.
It shows a decreasing of thrust 22 N/ m/s and with 355 kg, this gives about Xu = 0.06

Last edited by Jean Claude; 01-11-2017 at 07:01 AM.

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