V-n Diagram

Maneuver Diagram

This diagram illustrates the variation in load factor with airspeed for maneuvers. At low speeds the maximum load factor is constrained by aircraft maximum CL. At higher speeds the maneuver load factor may be restricted as specified by FAR Part 25.

The maximum maneuver load factor is usually +2.5 . If the airplane weighs less than 50,000 lbs., however, the load factor must be given by: n= 2.1 + 24,000 / (W+10,000)
n need not be greater than 3.8. This is the required maneuver load factor at all speeds up to Vc, unless the maximum achievable load factor is limited by stall.

The negative value of n is -1.0 at speeds up to Vc decreasing linearly to 0 at VD .
Maximum elevator deflection at VA and pitch rates from VA to VD must also be considered.

Gust Diagram

Loads associated with vertical gusts must also be evaluated over the range of speeds.
The FAR's describe the calculation of these loads in some detail. Here is a summary of the method for constructing the V-n diagram. Because some of the speeds (e.g. VB) are determined by the gust loads, the process may be iterative. Be careful to consider the alternative specifications for speeds such as VB.

The gust load may be computed from the expression given in FAR Part 25. This formula is the result of considering a vertical gust of specified speed and computing the resulting change in lift. The associated incremental load factor is then multiplied by a load alleviation factor that accounts primarily for the aircraft dynamics in a gust.

with: a = (dCL/da)
Ue = equivalent gust velocity (in ft/sec)
Ve = equivalent airspeed (in knots)
Kg = gust alleviation factor

Note that c is the mean geometric chord here.

The FAA specifies the magnitude of the gusts to be used as a function of altitude and speed:
Gust velocities at 20,000 ft and below:
66 ft/sec at VB
50 ft/sec at VC
25 ft/sec at VD.

Gust velocities at 50,000 ft and above:
38 ft/sec at VB
25 ft/sec at VC
12.5 ft/sec at VD.

These velocities are specified as equivalent airspeeds and are linearly interpolated between 20000 and 50000 ft.

So, to construct the V-n diagram at a particular aircraft weight and altitude, we start with the maximum achievable load factor curve from the maneuver diagram. We then vary the airspeed and compute the gust load factor associated with the VB gust intensity. The intersection of these two lines defines the velocity VB. Well, almost. As noted in the section on design airspeeds, if the product of the 1-g stall speed, Vs1 and the square root of the gust load factor at VC (ng) is less than VB as computed above, we can set VB = Vs1 sqrt(ng) and use the maximum achievable load at this lower airspeed.

Next we compute the gust load factor at VC and VD from the FAA formula, using the appropriate gust velocities. A straight line is then drawn from the VB point to the points at VC and VD.

Additional Notes on Computations

1) The lift curve slope may be computed from the DATCOM expression:

where b is the Prandtl-Glauert factor: b = sqrt(1-M2)

and k is an empirical correction factor that accounts for section lift curve slopes different from 2p. In practice k is approximately 0.97. This expression provides a reasonably good low-speed lift curve slope even for low aspect ratio wings. The effect is an important one as can be seen from the data for a DC-9 shown below. The maximum lift curve slope is about 50% greater than its value at low Mach numbers.

2) Recall CLmax may vary with Mach number as discussed in the high-lift section.

Details in FAR 25, not included here:

Check at all altitudes, weights, loading distributions.
Include pitching rates and pitch accelerations (dq/dt):
maximum elevator deflection at VA
Checked maneuver with dq/dt = 39 n (n-1.5) / V rad/sec2 or lower if not possible
For loads use this dq/dt at speeds from VA to VD combined with 1-g loads
also check: dq/dt = -29 n (n-1.5) / V combined with the positive maneuver load from VA - VD
Tail load due to gust can include full downwash and Kg-factor.