Download Engineering Problem Set: Recursive Binary Search, Fluid Dynamics, Signals and Systems and more Exercises Engineering in PDF only on Docsity!
CP11-
The problems in this problem set cover lectures C11 and C
a. Define a recursive binary search algorithm.
b. Implement your algorithm as an Ada95 program.
c. What is the recurrence equation that represents the computation time of your algorithm?
d. What is the Big-O complexity of your algorithm? Show all the steps in the computation based on your algorithm.
Turn in a hard copy of your algorithm, recurrence equation, and Big-O analysis, and code listing, and an electronic copy of your code.
- What is the Big-O complexity of: a. Heapify function b. Build_Heap function c. Heap_Sort
Show all the steps in the computation of the Big-O complexity.
Note: the code for heap_sort, build_heap and heapify was shown in lecture C11 and has been distributed via email.
Unified Engineering Spring Term 2004
Problem P3. (Propulsion) (LO A & B)
An incompressible fluid flows steadily through a two-dimensional
infinite row of fixed airfoils (e.g. a stator blade row). The blade
row is contained in a constant area annulus, as shown on the right
side of the figure below. The spacing between the airfoils is s.
Assume that the velocities and pressures Va, Vb, pa, pb, are
constant at stations (a) and (b), and that the flow angles are given
by βa and βb.
a) Does the magnitude of the flow velocity increase/stay the
same/decrease across the stator and why?
b) Using the control volume shown above (the upper and lower
surfaces are streamlines), apply conservation of mass and
momentum to determine the forces Rx and Ry that must be applied
to the fluid (these are equal and opposite to the forces needed to
keep each vane in place).
Unified Engineering II Spring 2004
Problem S10 (Signals and Systems)
This problem provides lots of practice using partial fraction expansions to determine inverse Laplace transforms. Please use the coverup method — it really is superior to other methods, and more reliable. Also, please check your answer, that is, verify that your expansion really is equivalent to the G(s) given.
For each of the following Laplace transforms, find the inverse Laplace transform.
3 s^2 + 3s − 10
- G(s) = , Re[s] > 2 s^2 − 4
6 s^2 + 26s + 26
- G(s) = , Re[s] > − 1 (s + 1)(s + 2)(s + 3)
4 s^2 + 11s + 9
- G(s) = (s + 1)^2 (s + 2)
, Re[s] > − 1
4 s^3 + 11s^2 + 5s + 2
- G(s) = , Re[s] > 0 s^2 (s + 1)^2
s^3 + 3s^2 + 9s + 12
- G(s) = , Re[s] > 0 (s^2 + 4) (s^2 + 9)
Unified Engineering II Spring 2004
Problem S11 (Signals and Systems)
Consider an aircraft flying in cruise at 250 knots, so that
v 0 = 129 m/s
Assume that the aircraft has lifttodrag ratio
L 0 = 15 D 0
Then the transfer function from changes in thrust to changes in altitude is
2 g 1 G(s) = (1) mv 0 s (s^2 + 2ζωns + ω^2 n)
where the natural frequency of the phugoid mode is
g ωn =
v 0
the damping ratio is
1 ζ = √ (3) 2(L 0 /D 0 )
and g = 9.82 m/s is the acceleration due to gravity. The transfer function can be normalized by the constant factor 2 g mv 0 , so that
¯^1 G(s) = (4) s (s^2 + 2ζωns + ω^2 n)
is the normalized transfer function, corresponding to normalized input
2 g u(t) = δT mv 0
- Find and plot the impulse response corresponding to the transfer function G^ ¯(s), using partial fraction expansion and inverse Laplace techniques. Hint: The poles of the system are complex, so you will have to do complex arithmetic.
- Suppose we try to control the altitude through a feedback loop, as shown below
+ (^) G(s)
r(t) e(t) u(t) h(t) k
Unified Engineering II Spring 2004
Problem S12 (Signals and Systems)
For each signal below, find the bilateral Laplace transform (including the region of convergence) by directly evaluating the Laplace transform integral. If the signal does not have a transform, say so.
- g(t) = sin(at)σ(−t)
- g(t) = teatσ(−t)
- g(t) = cos(ω 0 t) e−a| |^ t, for all t
