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Unified Engineering Fall 2003 Fluids Problems F3–F
F3. Anderson Problem 1.
F4. A rectangular wing of chord c and span b is operating in low-speed flow. Its drag depends on the following parameters:
D = f (�, � , V�, μ , b, c)
C
Determine the dimensionless parameters (Pi products) which determine the drag coefficient D.
F5. An certain aircraft normally operates at 8km altitude, where the air properties compare to the sea level values as follows:
� = 0. 50 �SL a = 0. 95 aSL μ = 0. 95 μSL
a) A 1/4 scale model of an aircraft is to be tested in an wind tunnel at sea level conditions. Is it possible to match both the Reynolds number and Mach number to those of the actual aircraft at altitude? Explain why or why not.
b) The Wright Brothers Wind Tunnel was designed to be pressurized during operation, as a means of increasing the air density �. The air temperature in the tunnel is still maintained at a normal 300 K, so that the speed of sound a and air viscosity μ inside the tunnel are unaffected by the pressurization, and are equal to their normal sea level values. Determine the pressure (in atmospheres) at which the tunnel must be operated to match the flight Reynolds and Mach numbers.
Unified Engineering I Fall 2003
Problem S12 (Signals and Systems)
The longitudinal dynamics of an aircraft flying are given in statespace form as
dh dt
= V sin γ (1) dV T D dt
= −g sin γ + m
m
dγ dt
L − mg mV
where
h = altitude V = velocity of aircraft γ = flight path angle g = acceleration due to gravity L = lift T = thrust D = drag m = mass of aircraft
We assume that (1) the thrust is constant, (2) the aircraft flies at constant coeeficient of lift, and (3) the aircraft flies at constant drag coeeficient. These assumptions are not bad for the case whent he pilot releases all the controls, and makes no adjustments, say, for altitude variations. In this case, we can rewrite the equations as
dh = V sin γ (4) dt dV T D 0 V 2 = −g sin γ + (5) dt m
m V 02 dγ L 0 (V /V 0 )^2 − mg = (6) dt mV
where L 0 and D 0 are the lift and drag at the nominal velocity, V 0. If the aircraft is intially in trim, it must also be true that
T = D 0 (7) L 0 = mg (8)
Equations (4)–(6) are the equations of motion in statespace form, although the equations are nonlinear. The equations may be linearized, by considering small per turbations δh, δV , and δγ about the nominal trajectory. The linearized equations of
Unified Engineering I Fall 2003
Problem S13 (Signals and Systems)
Consider the RLC circuit below:
u(t) + R y(t) C 2 3
L 1
This circuit is a notch filter, meaning that the output y(t) is almost the same as the input u(t), except that the circuit “filters out” frequencies in a narrow range, deter mined by the component values. For example, this circuit might be used to filter out 60 Hz noise caused by electrical wiring from the input to an audio system, to prevent 60 Hz “hum.”
For this circuit, find a statespace description of the system, in the form
dx(t) = Ax(t) + Bu(t) dt y(t) = Cx(t) + Du(t)
No component values are given, so just find the matrices A, B, C, and D in symbolic form.
Unified Engineering I Fall 2003
Problem S14 (Signals and Systems)
Consider the RLC circuit of Problem S13, shown below:
u(t) + R y(t) C 2 3
L 1
- Find the transfer function, G(s), of the system, using
G(s) = C(sI − A)−^1 B + D (1)
- Find the transfer function using impedance methods. Show that your result agrees with the result in part (1).
- For component values
L 1 = 1 H, C 2 = 0.25 F, R 3 = 10 Ω
plot the magnitude of the transfer function G(jω) vs. ω. Explain why the filter is called a notch filter.
Note: You may find it useful to use Matlab or a spreadsheet to calculate values of the transfer function, since there is a fair amount of complex arithmetic.
