It has been shown (Pounds, 2011) that an unloaded UAV helicopter is closed-loop stable and will have a characteristic equation given by mgh I mgh mgh +k= ·S+ (kki+q₁) = 0 I I + (918) ₁² m'gh' I' (92+kka) + where m is the mass of the helicopter, g is the gravitational constant, I is the rotational inertia of the helicopter, h is the height of the rotor plane above the center of gravity, 91 and 92 are stabilizer flapping parameters, k, kį, and ka are controller parameters; all constants > 0. The UAV is supposed to pick up a payload; when this occurs, the mass, height, and inertia change to m', h', and I', respec- tively, all still > 0. Show that the helicopter will remain stable as long as 2 - q₁+kki – 918k k(9₂+kka)
It has been shown (Pounds, 2011) that an unloaded UAV helicopter is closed-loop stable and will have a characteristic equation given by mgh I mgh mgh +k= ·S+ (kki+q₁) = 0 I I + (918) ₁² m'gh' I' (92+kka) + where m is the mass of the helicopter, g is the gravitational constant, I is the rotational inertia of the helicopter, h is the height of the rotor plane above the center of gravity, 91 and 92 are stabilizer flapping parameters, k, kį, and ka are controller parameters; all constants > 0. The UAV is supposed to pick up a payload; when this occurs, the mass, height, and inertia change to m', h', and I', respec- tively, all still > 0. Show that the helicopter will remain stable as long as 2 - q₁+kki – 918k k(9₂+kka)
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![**UAV Helicopter Stability Analysis**
It has been shown (Pounds, 2011) that an unloaded UAV helicopter is closed-loop stable and will have a characteristic equation given by:
\[
s^3 + \left(\frac{mgh}{I} (q_2 + kkd) + q_{1g}\right) s^2 + \frac{k \, mgh}{I} s + \frac{mgh}{I} (kki + q_1) = 0
\]
Where:
- \( m \) is the mass of the helicopter.
- \( g \) is the gravitational constant.
- \( I \) is the rotational inertia of the helicopter.
- \( h \) is the height of the rotor plane above the center of gravity.
- \( q_1 \) and \( q_2 \) are stabilizer flapping parameters.
- \( k, \, k_i, \) and \( k_d \) are controller parameters.
All constants are greater than 0. The UAV is supposed to pick up a payload; when this occurs, the mass, height, and inertia change to \( m', \, h', \) and \( I' \), respectively, all still greater than 0.
**Objective:**
Show that the helicopter will remain stable as long as:
\[
\frac{m'gh'}{I'} > \frac{q_1 + kki - q_{1g}k}{k(q_2 + kkd)}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F811e9e09-f827-40da-a830-8949fa4f6658%2Fb91941c4-9ef1-4e64-9ab0-d2925ed01547%2F5kro8sx_processed.png&w=3840&q=75)
Transcribed Image Text:**UAV Helicopter Stability Analysis**
It has been shown (Pounds, 2011) that an unloaded UAV helicopter is closed-loop stable and will have a characteristic equation given by:
\[
s^3 + \left(\frac{mgh}{I} (q_2 + kkd) + q_{1g}\right) s^2 + \frac{k \, mgh}{I} s + \frac{mgh}{I} (kki + q_1) = 0
\]
Where:
- \( m \) is the mass of the helicopter.
- \( g \) is the gravitational constant.
- \( I \) is the rotational inertia of the helicopter.
- \( h \) is the height of the rotor plane above the center of gravity.
- \( q_1 \) and \( q_2 \) are stabilizer flapping parameters.
- \( k, \, k_i, \) and \( k_d \) are controller parameters.
All constants are greater than 0. The UAV is supposed to pick up a payload; when this occurs, the mass, height, and inertia change to \( m', \, h', \) and \( I' \), respectively, all still greater than 0.
**Objective:**
Show that the helicopter will remain stable as long as:
\[
\frac{m'gh'}{I'} > \frac{q_1 + kki - q_{1g}k}{k(q_2 + kkd)}
\]
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