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Unformatted text preview: ME365 Homework Set 6 Out: September 29, 2011
Due: October 14, 2011 Problem No. 1
Given:
A shock absorber is described by the following differential equation: where m is the mass of the shock absorber, b is the damping coefficient of the
dashpot, k is the spring constant, K is the static gain, x is the displacement input (the
depth of a pothole or the height of a speed bump or curb) and y is the displacement
output of the shock absorber (the displacement the occupant of the vehicle feels).
For a particular shock absorber, m = 1250kg, k = 5000 N/m and K = 5000 N/m.
Find:
(a) Determine the natural frequency, , and damping ratio, , of this system
symbolically. Find the value of the damping constant, b, needed to attain a
damping ratio of 0.7.
(b) Using your value from (a), find the total time response of the shock absorber
(particular solution +homogeneous solution) for a pothole of depth 10 cm.
Assume that the shock absorbed starts from rest and that the pothole is a
step input.
(c) A batch of shock absorbers has been assembled with the wrong dashpot. The
actual damping coefficient is 1000 kg/s. At what frequency would a
sinusoidal input generate the largest cabin displacement? ME365 Homework Set 6 Out: September 29, 2011
Due: October 14, 2011 (a)
√ 2√ 2 ⋅ 0.7 ⋅ √5000 ⋅ 1250 ⋅ 3500 (b)
0⇒
5000 /
5000 /
Underdamped response ⇒ From ICs,
0 0 0 1.4 ⋅ 1 ⋅ ⋅1
0.1 0.1 1 , ⋅0 ⋅1 (c) ⋅0 → 0.1 1.428 0.14 ⋅1 →
0.0980 1000
2√
12 1.428 0.1 ⋅0 1⋅
. 1.400 . 0.1
0
0
0 0.1 1 ⋅ 0, 0.1 2√5000 ⋅ 1250
2 1 2 ⋅ 0.200 1.428 0.200
1.9183 / 0.0980 ME365 Homework Set 6 Out: September 29, 2011
Due: October 14, 2011 Problem No. 2
Given:
An unknown RC circuit in the laboratory has been experimentally determined to
have the frequency response given by the Bode pl ot below. You have been tasked
l
with identification of the circuit. The input to the c ircuit is a voltage at Vin, the output
of the circuit is a voltage at Vout. Bode Diagram
20 Magnitude (dB) 10 0 10 Phase (deg) 20
0 45 90
2 10 1 10 0 10 1 10 2 10 Frequency (rad/sec) Find:
(a) From your previous experience, you remember that an RC circuit can be
represented by a first order ODE. Use Ohm’s Law and Kirchoff’s Current Law
to determine the first order ODE. Use symbols, not values.
(b) Use the following measured frequency resp onse data a the figure above to
and
determine the static sensitivity and time co nstant. If the resistor was
measured to be R = 1 kΩ, what is the value of C?
 8  1.87
∠8
82.9 °
(c) What is the frequency response of the syste to the following input?
em
1 ∙ cos
What is the frequency response of the syste m to the following input?
1 ∙ cos 2 2
20 ME365 Homework Set 6 Out: September 29, 2011
Due: October 14, 2011 (a) (b)
∠ 8
 (c) tan
8 8 8 1 1
82.9° →
10 . → 1
10 10
17
1
11
∠1
tan 1 ⋅ 1
45°
10
 2 20 
21.85
11
1
∠ 2 20
tan 1 ⋅ 2 20
89.54°
 1 ME365 Homework Set 6 Out: September 29, 2011
Due: October 14, 2011 Problem No. 3
Given:
A piezoelectric stack actuator is a compact, high bandwidth device capable of
producing large forces but very small displacements. Many thin wafers of
piezoelectric material are sandwiched between electrodes and stacked together.
The result is that the wafers are electrically in parallel while mechanically in series.
Find:
(a) A firstcut model of the extension (in µm) of a piezoelectric stack actuator in
response to an input voltage is given by
2
where K = 0.18 µm/V, ωn = 3 kHz and ζ = 0.1. Determine the magnitude and
phase of the model frequency response as a function of the frequency, ω.
(b) Using your answer from (a), draw the Bode plot of the system model using
straight line approximations. Identify the magnitude and phase values at the
resonant frequency on your plot.
(c) Assume the extension of the piezoelectric stack actuator to be measured with
a sensor having a static sensitivity of 0.5 V/µm. For data collection, a firstorder lowpass filter is to be included between the displacement sensor and
the ADC. Two filters are available, having cutoff frequencies of 1 kHz or 10
kHz. Which filter is a better choice and why? For your selection, determine
the frequency response function of the combined piezoelectric stack
actuator, sensor and low pass filter. Draw the Bode plot of the combined
piezoelectric stack actuator, sensor and low pass filter using straight line
approximation.
1 ME365 Homework Set 6 Out: September 29, 2011
Due: October 14, 2011 (a)
 0.18 
2 2 ⋅ 0.1 ⋅
2 ⋅ 3000 1
2 ∠ tan 4 ⋅ 3000 2 ⋅ 0.1 ⋅
2 ⋅ 3000 tan
1 1 1 4 ⋅ 3000 (b) Drawing will be a straight line approximation, should roughly match the plot
given in this solution.
Bode Diagram
0 Magnitude (dB) 20
40
60
80
 100
0 Phase (deg) 45
90
135
180
3 4 10 5 10 6 10 10 Frequency (rad/sec) 1 2 2 ⋅ 3000 1 2 ⋅ 0.1 
∠ 0.8715
84.2318°  18660 2970 (c) 10 kHz is better choice to not roll off the peak. Drawing will be a straight line
approximation, should roughly match one of the plots given in this solution. ME365 Homework Set 6 Bode Diagram Original System 0 Original System + 1 kHz LP 50
Magnitude (dB) Out: September 29, 2011
Due: October 14, 2011 Original System + 10 kHz LP
100 150 Phase (deg)  200
0 90 180 270
2 10 3 10 4 10 5 10 Frequency (rad/sec) 6 10 7 10 ...
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