V mg Sw mg 2. A drone of mass m (shown above) moves with a horizontal velocity v and is propelled by a thrust force ft produced from its propellers. To move, it tilts itself by an angle 0, which causes the thrust force ft to take on a component in the horizontal direction. This forward-directed force is indicated by ff in the diagram above. The drone is also subject to an aerodynamic drag force fa that acts in a horizontal direction and is proportional to forward velocity according to fd = -bu (regardless of 0). (a) Assuming ft acts along the inclined axis of the body, find ff as a function of the tilt angle and ft. (b) Find the necessary thrust ft needed to ensure that lift is balanced by gravity so that the drone does not move vertically. That is, find ft such that its vertical component as shown is equal to mg. Your answer will be a function of 0, g, and m. (c) Give the the small-angle approximations of these two functions when 0≈ 0. (d) Suppose the drone has a mass m = 1 kg and damping coefficient b = 0.2 N/m, and is at rest. At time t= 0, it tilts itself (assume this happens instantaneously) such that the forward force ff is a constant 0.5 N. i. Give a differential equation describing the horizontal dynamics of this system. ii. Solve the differential equation to give the horizontal velocity v(t) of the drone and sketch the function, showing its initial value, slope, and asymptotic value if any. iii. What is its time constant, if any? iv. Is this a step response or an impulse response? = Now suppose instead that it is stationary and upright (0 0) and it is subject to a sudden gust of wind that is well-modeled as an impulse force fw (t) = 18(t), where the impulsive is I = 3 Ns. i. Find and sketch v(t) under these conditions. ii. How much time does it take before the drone's velocity has reduced to 1/10th of the velocity immediately after the impulse?
V mg Sw mg 2. A drone of mass m (shown above) moves with a horizontal velocity v and is propelled by a thrust force ft produced from its propellers. To move, it tilts itself by an angle 0, which causes the thrust force ft to take on a component in the horizontal direction. This forward-directed force is indicated by ff in the diagram above. The drone is also subject to an aerodynamic drag force fa that acts in a horizontal direction and is proportional to forward velocity according to fd = -bu (regardless of 0). (a) Assuming ft acts along the inclined axis of the body, find ff as a function of the tilt angle and ft. (b) Find the necessary thrust ft needed to ensure that lift is balanced by gravity so that the drone does not move vertically. That is, find ft such that its vertical component as shown is equal to mg. Your answer will be a function of 0, g, and m. (c) Give the the small-angle approximations of these two functions when 0≈ 0. (d) Suppose the drone has a mass m = 1 kg and damping coefficient b = 0.2 N/m, and is at rest. At time t= 0, it tilts itself (assume this happens instantaneously) such that the forward force ff is a constant 0.5 N. i. Give a differential equation describing the horizontal dynamics of this system. ii. Solve the differential equation to give the horizontal velocity v(t) of the drone and sketch the function, showing its initial value, slope, and asymptotic value if any. iii. What is its time constant, if any? iv. Is this a step response or an impulse response? = Now suppose instead that it is stationary and upright (0 0) and it is subject to a sudden gust of wind that is well-modeled as an impulse force fw (t) = 18(t), where the impulsive is I = 3 Ns. i. Find and sketch v(t) under these conditions. ii. How much time does it take before the drone's velocity has reduced to 1/10th of the velocity immediately after the impulse?
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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Question
Transcribed Image Text:=↑
-7-----
mg
Sw
mg
2.
A drone of mass m (shown above) moves with a horizontal velocity v and is propelled by
a thrust force ft produced from its propellers. To move, it tilts itself by an angle 0, which causes
the thrust force ft to take on a component in the horizontal direction. This forward-directed force is
indicated by ff in the diagram above. The drone is also subject to an aerodynamic drag force fa that
acts in a horizontal direction and is proportional to forward velocity according to fa= -bv (regardless
of 0).
(a) Assuming ft acts along the inclined axis of the body, find ff as a function of the tilt angle and
ft.
(b) Find the necessary thrust ft needed to ensure that lift is balanced by gravity so that the drone
does not move vertically. That is, find ft such that its vertical component as shown is equal to
mg. Your answer will be a function of 0, g, and m.
(c) Give the the small-angle approximations of these two functions when ≈ 0.
(d) Suppose the drone has a mass m = 1 kg and damping coefficient b = 0.2 N/m, and is at rest. At
time t = 0, it tilts itself (assume this happens instantaneously) such that the forward force ff is
a constant 0.5 N.
i. Give a differential equation describing the horizontal dynamics of this system.
ii. Solve the differential equation to give the horizontal velocity v(t) of the drone and sketch the
function, showing its initial value, slope, and asymptotic value if any.
iii. What is its time constant, if any?
iv. Is this a step response or an impulse response?
(e) Now suppose instead that it is stationary and upright (0 =0) and it is subject to a sudden gust
of wind that is well-modeled as an impulse force fw (t) = 18(t), where the impulsive is I = 3 Ns.
i. Find and sketch v(t) under these conditions.
ii. How much time does it take before the drone's velocity has reduced to 1/10th of the velocity
immediately after the impulse?
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