Description
Consider a flight formation scenario where the jets can only talk with its adjacent neighbors, as indicated by the blue lines below.
In this scenario, the ππth jet adjusts its velocity, i.e., π£π£ππ(π‘π‘), on the direction of π¦π¦-axis as
πππ£π£ππ(π‘π‘)
ππππ = β 1
|ππππ|
οΏ½ πΌπΌοΏ½π¦π¦ππ(π‘π‘) β π¦π¦ππ (π‘π‘) β Ξππππ οΏ½ + π½π½ οΏ½π£π£ππ(π‘π‘) β π£π£ππ (π‘π‘)οΏ½
ππβππππ
, (1)
where πΌπΌ and π½π½ are the stiffness and damping coefficients, respectively, π¦π¦ππ(π‘π‘) is the position of the ππth jet on the y-axis, and it can be expressed as
π¦π¦ππ(π‘π‘) = π¦π¦ππ(0) + οΏ½ π£π£ππ(π‘π‘)
π‘π‘
0
ππππ, (2)
ππππ is the set of neighbor indices of the ππth jet, |ππππ| is the cardinality of the set ππππ (i.e., the number of neighbors of the ππth jet), and Ξππππ is the
desired distance between the ππth jet and the ππth jet for π‘π‘ β β. For example, for this scenario, Ξ12 β limπ‘π‘ββ π¦π¦1(π‘π‘) β π¦π¦2(π‘π‘) = βππ and Ξ21 β
limπ‘π‘ββ π¦π¦2(π‘π‘) β π¦π¦1(π‘π‘) = ππ. (Please pay attention to the signs in your expressions.)
For the initial positions οΏ½π¦π¦1(0), π¦π¦2(0), π¦π¦3(0),π¦π¦4(0), π¦π¦5(0)οΏ½ = (0,20,40,60,80), initial velocities οΏ½π£π£1(0), π£π£2(0), π£π£3(0), π£π£4(0), π£π£5(0)οΏ½ =
(500,500,500,500,500), πΌπΌ = 1, π½π½ = 2, and ππ = 10,
1. (25 pts) Determine limπ‘π‘ββπ£π£ππ(π‘π‘) with MATLAB for all ππ
2. (25 pts) Determine ππππ(π π ) with MATLAB for all ππ
3. (25 pts) With WolframAlpha, calculate the inverse Laplace transform of ππ3(π π ) and plot π£π£3(π‘π‘) in MATLAB
4. (25 pts) By using the approximation πππ£π£ππ(π‘π‘)
ππππ β π£π£ππ(π‘π‘+Ξπ‘π‘)βπ£π£ππ(π‘π‘)
Ξπ‘π‘ in (1),
o Develop a MATLAB code that obtains π£π£ππ(π‘π‘) numerically for π‘π‘ β [0,20] seconds for ππ (Hint: Choose Ξπ‘π‘ = 0.001 and use it
in (1) and (2))
o Plot π£π£ππ(π‘π‘) for all the jets (Hint: π£π£3(π‘π‘) should match with the result in part 3)
o Plot π¦π¦2(π‘π‘) β π¦π¦1(π‘π‘), π¦π¦3(π‘π‘) β π¦π¦2(π‘π‘), π¦π¦3(π‘π‘) β π¦π¦4(π‘π‘), and π¦π¦4(π‘π‘) β π¦π¦5(π‘π‘) (Hint: They should approach ππ = 10 as π‘π‘ β β)