ENGM2022 - Laplace Transforms And nth Derivative - Management Assignment Help

Assignment Task

1.  Consider the Newton cooling model, dT/dt + kT = kT∞(t), for tem- perature T(t) of an object with rate constant k subject to a piecewise external temperature T∞(t): T∞(0) = 30, T∞(2) = 25, T∞(4) = 30, and T∞(t) = 30 = thereafter. Assume that T(0) = 20.
 

(a) Find a formula for T∞(t) and find its laplace transform. To find T∞(t) find the equations of the lines between 0 ≤ t ≤ 2 and 2 ≤ t ≤ 4 and then use our

window function to create T∞(t). You should find that

(b)  Assume that L(T(t)) = W(s) and use our 4-step method to find T(t). Label each step. In your solution use the definitions: F1(s) =

1/(s + k), F2(s) = 1/(s(s + k)), F3(s) = 1/(s 2 (s + k)). In step 3 be sure to verify that fi(0) = 0, i = 2, 3. You should find that

(c) In order for the temperature T to approach T = 30 as t → ∞ all of the constant terms must sum to 30 and the linear terms in ’t’ must sum to zero.

Show that this occurs in your solution. Assume that t ≥ 4 so that the Heaviside step terms are all U=1.

2.  Consider the critically-damped system  ?y + 2  ?y + y = F(t), with zero initial conditions, where F(t) is the function F(t) = 1 − t 2 switched on from t = 3 to t = 4.

(a) Find a formula for F(t) and find its laplace transform. To find F(t) use our windowed approach to switch on f(t) from t = 3 to t = 4 by first shifting f(t)

ahead by 3 time units and then turning that function ON at t = 3 and OFF at = 4. You should find that

(b) Use our 4-step method to find y(t) and assume that the laplace transform of y(t) is Y (s). Label each step. In your solution use the defi- nitions:

Fi(s) = 1/[s i (s + 1)2 ], i = 1, 2, 3. In step 3 be sure to verify that  fi(0) = 0,  ?fi(0) = 0, i = 1, 2, 3. You should find that.

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