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5 Questions total

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ENGM2022: Assignment 3: Winter Term 2021
how solutions are to be constructed. Use our 4-step MUD (Method of
Undetermined Coefficients) for inhomogeneous problems after problem 1
– Step 1: Get homogeneous solution yh , Step 2: Get particular solution
yp , Step 3: Satisfy conditions on y = yh + yp , Step 4: Sketch y. Show your
work. The sketch featues always need to be justified and this is spelled
out in each problem as clearly as possible.
1. (20 points) Find the solution of yÃŒË† + 2cyÃŒâ€¡ + 2y = 0, y(0) = y0 , yÃŒâ€¡(0) = v0 . In your
solution clearly label the following steps: Step 1: Characteristic Equation, Step
2: General Solution, Step 3: Satisfy conditions, Step 4: Sketch.
(a) (1 pt) Choose c > 0 so that the model is critically damped.
(b) (3 pt) Solve the critically-damped model, but do not sketch, for y(0) = y0 ,
yÃŒâ€¡(0) = v0 and show that
y(t) = (y0 + (v0 + cy0 )t)) eÃ¢Ë†â€™ct
(1)
(c) (3 pt) Now sketch critically-damped y(t) assuming that y0 = 1 and v0 =
Ã¢Ë†â€™2. In your sketch youÃ¢â‚¬â„¢ll need to determine if y can go below y = 0. To
receive any sketch points use pure logic without calculus to show whether y
goes below zero. Complete the sketch with that single piece of information
– and the facts we outlined in class about the number of zero crossings
available to damped solutions of second order differential equations.
(d) (4 pt) Assume that c is large enough to make the model over-damped.
Find the solution y(t) and do not sketch. Show that
(y0 ÃŽÂ»2 Ã¢Ë†â€™ v0 )eÃŽÂ»1 t + (v0 Ã¢Ë†â€™ y0 ÃŽÂ»1 )eÃŽÂ»2 t
ÃŽÂ»2 Ã¢Ë†â€™ ÃŽÂ»1
Ã¢Ë†Å¡
Ã¢Ë†Å¡
where ÃŽÂ»1 = Ã¢Ë†â€™c + c2 Ã¢Ë†â€™ 2 and ÃŽÂ»2 = Ã¢Ë†â€™c Ã¢Ë†â€™ c2 Ã¢Ë†â€™ 2.
y=
(2)
(e) (2 pt) Show that the over-damped solution satisfies the conditions y(0) =
y0 , yÃŒâ€¡(0) = v0 .
(f) (1 pt) Refer to the over-damped solution and find a formula to choose y0
and v0 so that the solution goes to zero as rapidly as possible – this will
require the coefficient of the slowest part of y to be zero.
(g) (6 pt) Assume that c is small enough to make the model under-damped.
Show that
q
y(t) = eÃ¢Ë†â€™ct y02 + ((v0 + cy0 )/Ãâ€°)2 cos(Ãâ€°t Ã¢Ë†â€™ Ãâ€ )
(3)
Ã¢Ë†Å¡
where Ãâ€° = 2 Ã¢Ë†â€™ c2 and tan Ãâ€  = ((v0 + cy0 )/Ãâ€°)/y0 . In your sketch please
draw the solution envelope and identify the psuedo-period.
2. (10 points) Consider Problem #4 Assignment 2. Assume that the person in the
question is not a woman being tested for gestational diabeters but is a child
that repeatedly asks for a sugary drinks in a periodic fashion in response to
Ã¢â‚¬â„¢hunger pangsÃ¢â‚¬â„¢. In this case, assume that GÃ¢Ë†Å¾ (t) = G1 cos(Ãâ€°t).
(a) (1 point) If the child asks for a sugary drink once per hour and we work
in hours as our time unit (e.g. not seconds) then what is Ãâ€°?
(b) (1 point) Just For Fun Ã¢â‚¬â€œ approximating is everywhere and very helpful.
There are about 30g of sugar in one 340ml (about 12oz – typical pop can
size) of typical orange juice. If a 22kg (about 50lb) child drinks two 340ml
cans of juice/day, each containing about 30g of sugar, in how many days
will they consume their body weight in sugar?
(c) (2 points) Write down and solve the auxiliary problem Y (t) in this problem.
Show that the auxiliary solution is
Y (t) =
eat
k+a
(4)
where a = iÃâ€°.
(d) (4 points) Follow the 4-step MUD and use the auxiliary solution Y (t) to
construct the particular solution in Step 2. Omit Step 4: Sketch. Show
that the solution satisfying the condition is
G(t) = k1 G1
k cos(Ãâ€°t) + Ãâ€° sin(Ãâ€°t)
+ AeÃ¢Ë†â€™kt
k2 + Ãâ€°2
(5)
with A = G0 Ã¢Ë†â€™ k1 kG1 /(k 2 + Ãâ€° 2 ).
(e) (2 points) Use the magnitude of the auxiliary problem solution to construct
the transfer function and sketch the transfer function.
3. (8 points) Refer to Problem #1 in this assignment. Consider the model in #1(a)
with external input F0 t exp(Ã¢Ë†â€™ct) and initial conditions: y(0) = 1, yÃŒâ€¡(0) = Ã¢Ë†â€™4.
Use the 4-step MUD method to solve this problem and omit Step 4: Sketch.
Step 1: (1 point), Step 2: (5 points), Step 3: (2 points), Step 4: omit. Show
that
y(t) = (E + F t)eÃ¢Ë†â€™ct +
F0 3 Ã¢Ë†â€™ct
te
6
(6)
Ã¢Ë†Å¡
where E = 1 and F = c Ã¢Ë†â€™ 4 and c = 2. Note: To receive full points in
Ã¢â‚¬â„¢Step 2 Get yp Ã¢â‚¬â„¢: (i) you will need to use our Ã¢â‚¬â„¢parallelÃ¢â‚¬â„¢ algebra technique that
you were shown during class – you will need this method in Laplace Transform
switching problems, (ii) clearly
show why the guess must be yp (t) = t2 (At +
Ã¢Ë†Å¡
B) exp(Ã¢Ë†â€™ct), where c = 2 to avoid duplication, and (iii) use our recursive
method of differentiating exponential * polynomial functions when finding yÃŒâ€¡p
and yÃŒË†p (the dot refers to d/dt and is a common notation that we also use in our
notes). All of these are fully explained in the Week5Class2 WedFeb3 video. It
will
Ã¢Ë†Å¡ be convenient, if you wish, to leave in the parameter Ã¢â‚¬â„¢cÃ¢â‚¬â„¢ instead of writing
2 and onceÃ¢Ë†Å¡the last step is complete in the parallel method you can replace
Ã¢â‚¬â„¢cÃ¢â‚¬â„¢ with c = 2 and the cancellations will occur as expected.
4. (5 points) Consider the problem in Case 3 underdamped, Week4Class3 FriJan29.
LetÃ¢â‚¬â„¢s apply an external input to that problem of the form A cos(ÃŽÂ±t) + B sin(ÃŽÂ²t).
(a) (1 point) Write down the auxiliary problem Y (t) and solve for Y (t) and
show that
eat
(7)
Y (t) = 2
a + 2a + 5
where a = iÃâ€° and we will be setting Ãâ€° = ÃŽÂ± and Ãâ€° = ÃŽÂ² later in this problem.
(b) (2 points) Use the auxiliary problem to construct the particular solution
yp (t) and show that
yp (t) =
A cos(ÃŽÂ²t Ã¢Ë†â€™ ÃŽÂ¦)
A cos(ÃŽÂ±t Ã¢Ë†â€™ Ãâ€ )
+
2
2
2
(5 Ã¢Ë†â€™ ÃŽÂ± ) + 4ÃŽÂ±
(5 Ã¢Ë†â€™ ÃŽÂ² 2 )2 + 4ÃŽÂ² 2
(8)
where tan(Ãâ€ ) = (2ÃŽÂ±)/(5 Ã¢Ë†â€™ ÃŽÂ±2 ) and tan(ÃŽÂ¦) = (5 Ã¢Ë†â€™ ÃŽÂ² 2 )/(Ã¢Ë†â€™2ÃŽÂ²)
(c) (1 point) Use the solution of the auxiliary problem to construct the transfer
function and note on your figure why this is sometimes referred to as a
Ã¢â‚¬â„¢band-passÃ¢â‚¬â„¢ amplifier. Use the variable Ãâ€° as the frequency of the external
input.
(d) (1 point) Use calculus to precisely location the input frequency associated
with the transfer function peak.
5. (7 points) Consider problem #1 of this assignment again. This time set the
damping c = 0 so that there is no damping (this is maximum under-damping
for this model).
Ã¢Ë†Å¡
(a) (4 points) Solve the problem for external input sin(Ãâ€°t) where Ãâ€° 6= 2.
Solve the problem using our 4-step MUD method and employ the auxiliary
problem Y (t) in finding the particular solution yp (t) in Step 2. Assume
zero initial conditions. Step 1: (1 point), Step 2: (2 points), Step 3: (1
point), Omit Step 4: Sketch. Show that

#
Ã¢Ë†Å¡
1
Ãâ€°
y(t) =
Ã¢Ë†â€™ Ã¢Ë†Å¡ sin( 2t) + sin(Ãâ€°t)
(9)
2 Ã¢Ë†â€™ Ãâ€°2
2
(b) (2 points)
Note that Ã¢Ë†Å¡
the solution is proportional to R(t) = sin(Ãâ€°t) Ã¢Ë†â€™
Ã¢Ë†Å¡
sin( 2t) when Ãâ€° Ã¢â€°Ë† 2. Use the trigonometry approach in the notes
to show that R(t) may
of the fast
Ã¢Ë†Å¡ be written as two times the product Ã¢Ë†Å¡
oscillation cos{(Ãâ€° + 2)t/2} and the slow oscillation sin{(Ãâ€° Ã¢Ë†â€™ 2)t/2}.
(c) (1 point) Sketch R(t) by using the result in (5b). Show the slow envelope
surrounding the fast oscillation and note the period of each on your sketch
as we did during class.