Principles of Flight

[Keith.Williams.]

Consider two conditions.
Condition 1 is straight and level flight in still air.
Let CL = CL1 and V = V1

Lift 1= CL1 1/2ρ V1squared S

In straight and level flight Lift = weight so we can say that
Weight = CL1 1/2ρ V1squared S

Rearranging gives
S = Weight / CL1 1/2ρ V1squared……………………Equation 1

Condition 2 is in a sudden vertical gust.
Let CL CL2 and V = V2

Lift 2 = CL2 1/2ρ V2squared S

Rearranging this gives
S = Lift 2/ CL2 1/2ρ V2squared …………………….Equation 2

S is the wing area which, for the same aircraft is the same in both cases. So we can combine equations 1 and 2 to give.

Lift 2/ CL2 1/2ρ V2squared = Weight / CL1 1/2ρ V1squared

Multiplying both sides by 1/2ρ gives
Lift 2/ CL2 V2squared = Weight / CL1 V1squared

Rearranging gives
Lift 2/ Weight = CL2 V2squared / CL1 V1squared

But Lift2 / Weight = new load factor so
New load factor = CL2 V2squared / CL1 V1squared

If we assume that V1 is VS then we have
New load factor = CL2 V2squared / CLmax Vs squared

If we also assume that the aircraft stalls in the gust then we have
New load factor = CLmax V2squared / CLmax Vs squared

Cancelling out Clmax from the top and bottom of the right side gives
New load factor = V2squared / Vs squared

Taking the figures Vs = 1 and V2 = 1.3 Vs = 1.3 gives
New load factor = 1.3 squared / 1 squared = 1.69
So the load factor in the gust is 1.69

BUT WE HAVE ASSUMED THAT BOTH AIRCRAFT STALLED.

If one or both of them did not stall then we cannot assume that both were at CLmax and so we cannot give a definite value for load factor. We can however say that the gust cannot (theoretically) produce a load factor greater than 1.69 when flying at 1.3vs.