Determine Vx and Vy from other climbing rate datas

Question

I am developing a small program, working with Microsoft Flight Simulator 2020. For now, I can show my takeoff graph which display the altitude on the y axis and the distance in x axis. I would like to add Vx and Vy curve on this chart but I can find an easy way to determine these curves with the data I have. So my question :

Based on my actual indicated speed and climbing rate. If I know that my Vx speed is, let's say 55kt, how can I find the corresponding climbing rate ?

More precisely, if (at full power) :

Vi=62kt => climbing rate is 750ft/min (data that I know)

So :

Vx = 55kt => climbing rate is ????

Vy = 70kt => climbing rate is ???

I know there is some formulas including weight, drag, etc.. but is there a way to find the climbing rate Vx, Vy with already a Vi/climbing rate relation that I know like above ? I'm sure there's a way to find it but I'm stuck. Thank you a lot for your information.

Answer 1

Since you have only one data point and no other information about the airplane, it is impossible to calculate the climb rates exactly. It seems your data point is neither at best climb angle nor best climb rate, but somewhere in between. Since the climb speed at both points is not so different, the climb speed at your in-between point should already be close to the desired values.

Now I have to guess: Your airplane is piston-powered, with a propeller. Next, I assume a constant speed propeller. The general behavior of thrust and drag over speed will then look like in the diagram below, which is for a generic propeller airplane. Change the numbers on the axes, and it will describe your airplane well:

The best climb angle is achieved close to stall speed and the highest rate of climb at half way to the point of best L/D. To see the formulas for calculating both speeds see this rsp. this answer. With your one data point and the shallow gradient of the climb rate over speed curve, I would expect that your best rate of climb is only a few percent better than the 750 ft/min at 62 kts and your best angle climb speed a few percent below. Since we are missing essential information like power setting, wing aspect ratio and zero-lift drag (fixed gear or not?), any further speculation will stand on very weak feet.

Answer 2

While there may be only a few degrees difference in angle of climb between Vx and Vy, let's see what those few degrees mean.

The easiest way to determine climb rate at Vx and Vy is to actually fly it.

However, fairly accurate mathematical approximations of rate of climb may be made if one knows the angle of climb and the airspeed, simply as sine $\theta$ × airspeed

If climb angle is unknown, one may derive an angle of climb by calculating excess thrust at Vx and Vy, and building a trigonometric function vs the weight of the aircraft. Note that there is more excess thrust at Vx, allowing for higher angle of climb, but the aircraft is aerodynamicly "cleaner" at Vy, allowing for more efficient use of thrust.

The result is a lower angle of climb at Vy, but a greater rate of climb.