Question
The following data are provided for this transport aircraft:
• Design dive speed VD = 293.39 [m/s].
• Lift curve slope dCL
dα = 5.0 [rad−1
].
• Maximum vertical gust velocity (U = 27.887 [ft/s]) to be considered at 4200 m altitude.
• Air density at 4200 m altitude: ρ = 0.8020 [kg/m3
].
• Aircraft weight W = 600 [kN].
• Wing surface area S = 120 [m2
].
B) Compute the total load factor for this aircraft when operating at 4200 m altitude. Note: you
already have a load factor of 1-g in level flight without a gust.
C) The vertical gust that should be considered at the design dive speed results in a sudden load
factor increase. When is this most critical from an operational perspective?
1) When flying at maximum aircraft weight
2) When flying at minimum aircraft weight
3) This limit is independent of the aircraft weight
• Design dive speed VD = 293.39 [m/s].
• Lift curve slope dCL
dα = 5.0 [rad−1
].
• Maximum vertical gust velocity (U = 27.887 [ft/s]) to be considered at 4200 m altitude.
• Air density at 4200 m altitude: ρ = 0.8020 [kg/m3
].
• Aircraft weight W = 600 [kN].
• Wing surface area S = 120 [m2
].
B) Compute the total load factor for this aircraft when operating at 4200 m altitude. Note: you
already have a load factor of 1-g in level flight without a gust.
C) The vertical gust that should be considered at the design dive speed results in a sudden load
factor increase. When is this most critical from an operational perspective?
1) When flying at maximum aircraft weight
2) When flying at minimum aircraft weight
3) This limit is independent of the aircraft weight
Answers
GPT-4o mini
To calculate the total load factor for the aircraft when operating at 4200 m altitude, we need to account for the effects of a vertical gust. The total load factor \( n \) can generally be expressed as:
\[
n = \frac{L}{W}
\]
where:
- \( L \) is the lift,
- \( W \) is the aircraft weight.
### Step A: Calculate Lift (L)
The lift can be calculated using the lift equation:
\[
L = \frac{1}{2} \rho V^2 S C_L
\]
To calculate \( C_L \), we first need to determine the angle of attack \( \alpha \) at the design dive speed \( V_D \):
From the information given, we know:
- Lift curve slope: \( \frac{dC_L}{d\alpha} = 5.0 \, \text{[rad}^{-1}] \)
Assuming a typical \( C_{L0} \) (lift coefficient at zero angle of attack) of 0 (for simplicity), the lift coefficient \( C_L \) at the design dive speed can be approximated as:
\[
C_L = \frac{dC_L}{d\alpha} \cdot \alpha
\]
To find the angle of attack \( \alpha \) for level flight, we use:
\[
W = L \implies W = \frac{1}{2} \rho V^2 S C_L
\]
### Step B: Calculate Load Factor in Level Flight
1. Weight \( W \):
\[
W = 600 \, \text{kN} = 600,000 \, \text{N}
\]
2. Air density \( \rho \):
\[
\rho = 0.8020 \, \text{kg/m}^3
\]
3. Wing surface area \( S \):
\[
S = 120 \, \text{m}^2
\]
4. Design dive speed \( V_D \):
\[
V_D = 293.39 \, \text{m/s}
\]
Now we can compute \( L \):
\[
\begin{align*}
L &= \frac{1}{2} \cdot 0.8020 \cdot (293.39)^2 \cdot 120 \cdot C_L
\end{align*}
\]
### Step C: Calculate Total Load Factor \( n \) Including Gust
In level flight without gusts, the load factor is:
\[
n_0 = 1 \implies n_0 = \frac{L}{W}
\]
When accounting for a vertical gust, the increase in lift due to the gust \( U \) is added to the lift:
The increment in lift due to the gust can be calculated by introducing the gust velocity \( U \):
\[
L' = \frac{1}{2} \rho (V_D + U)^2 S C_L
\]
Now, the additional load factor \( n_g \) caused by the gust can be calculated, and we add this to the original load factor:
\[
n' = \frac{L'}{W}
\]
### Timing of Critical Vertical Gust Load Factor
The criticality of vertical gusts in terms of load factor is influenced by the aircraft weight. The relationship generally holds that:
1. When flying at maximum aircraft weight, the total load factor increase due to gusts is more critical because the aircraft is closer to its structural limits.
2. When flying at minimum aircraft weight, the absolute load factors could be lower, but the relative increases may still matter.
**Conclusion**: The most critical situation from an operational perspective for load factor increases due to vertical gusts is typically:
**1) When flying at maximum aircraft weight**.
To summarize, at maximum weight, the aircraft is farthest away from structural design limits in terms of performance, and any additional loads (like those from gusts) take a larger percentage of the total allowable limits.
\[
n = \frac{L}{W}
\]
where:
- \( L \) is the lift,
- \( W \) is the aircraft weight.
### Step A: Calculate Lift (L)
The lift can be calculated using the lift equation:
\[
L = \frac{1}{2} \rho V^2 S C_L
\]
To calculate \( C_L \), we first need to determine the angle of attack \( \alpha \) at the design dive speed \( V_D \):
From the information given, we know:
- Lift curve slope: \( \frac{dC_L}{d\alpha} = 5.0 \, \text{[rad}^{-1}] \)
Assuming a typical \( C_{L0} \) (lift coefficient at zero angle of attack) of 0 (for simplicity), the lift coefficient \( C_L \) at the design dive speed can be approximated as:
\[
C_L = \frac{dC_L}{d\alpha} \cdot \alpha
\]
To find the angle of attack \( \alpha \) for level flight, we use:
\[
W = L \implies W = \frac{1}{2} \rho V^2 S C_L
\]
### Step B: Calculate Load Factor in Level Flight
1. Weight \( W \):
\[
W = 600 \, \text{kN} = 600,000 \, \text{N}
\]
2. Air density \( \rho \):
\[
\rho = 0.8020 \, \text{kg/m}^3
\]
3. Wing surface area \( S \):
\[
S = 120 \, \text{m}^2
\]
4. Design dive speed \( V_D \):
\[
V_D = 293.39 \, \text{m/s}
\]
Now we can compute \( L \):
\[
\begin{align*}
L &= \frac{1}{2} \cdot 0.8020 \cdot (293.39)^2 \cdot 120 \cdot C_L
\end{align*}
\]
### Step C: Calculate Total Load Factor \( n \) Including Gust
In level flight without gusts, the load factor is:
\[
n_0 = 1 \implies n_0 = \frac{L}{W}
\]
When accounting for a vertical gust, the increase in lift due to the gust \( U \) is added to the lift:
The increment in lift due to the gust can be calculated by introducing the gust velocity \( U \):
\[
L' = \frac{1}{2} \rho (V_D + U)^2 S C_L
\]
Now, the additional load factor \( n_g \) caused by the gust can be calculated, and we add this to the original load factor:
\[
n' = \frac{L'}{W}
\]
### Timing of Critical Vertical Gust Load Factor
The criticality of vertical gusts in terms of load factor is influenced by the aircraft weight. The relationship generally holds that:
1. When flying at maximum aircraft weight, the total load factor increase due to gusts is more critical because the aircraft is closer to its structural limits.
2. When flying at minimum aircraft weight, the absolute load factors could be lower, but the relative increases may still matter.
**Conclusion**: The most critical situation from an operational perspective for load factor increases due to vertical gusts is typically:
**1) When flying at maximum aircraft weight**.
To summarize, at maximum weight, the aircraft is farthest away from structural design limits in terms of performance, and any additional loads (like those from gusts) take a larger percentage of the total allowable limits.