ME 36500 EXAM #2 Tuesday, November 11, 2014

6:30 – 7:30 pm PHYS 112 and 114

Division: Chiu(10:30) / Shelton(11:30) / Bae(1:30) (circle one) HW ID: ____________

Name: ___________________________

Instructions

(1) This is a closed book examination, but you are allowed two single-sided 8.5”×11” crib sheet. (2) You have one hour to work all four problems in the exam. (3) Write your name and HW ID on the top of each page.

(4) Use the solution procedure: what are you given, what are you asked to find, what are your assumptions, what is your solution, does your solution make sense. You must show all of your work to receive any credit. Clearly mark up your answer.

(5) You must write neatly and should use a logical format to solve the problems. You are encouraged to really “think” about the problems before you start to solve them.

(6) If you use extra pages, make sure to write your NAME and HWID on the top. Make sure to sort the pages in the correct order and re-staple the packet together.

(7) Pay attention to units and remember to write down the units as needed.

(8) You are only allowed to use the ME authorized exam calculator, the TI-30XIIs. (9) You are not allowed to use your cellphone during exam. Please TURN OFF YOU CELLPHONE. Problem No. 1 (40 Points) ________________________ Problem No. 2 (40 Points) ________________________ Problem No. 3 (40 Points) ________________________ Problem No. 4 (30 Points) ________________________ TOTAL (***/150 Points) ________________________

Name: ____________________________ HW ID: _____________

Page  2  of  15  

Problem 1 (40 Points) A shock absorber is described as the following 2nd order differential equation:

40000!!y+2000 !y+100y = 200x

(A) (14 Points) Calculate the corresponding natural frequency, damping ratio, static sensitivity and resonance frequency.

40000100

!!y +2000100

!y + y =200100

x

400 !!y + 20 !y + y = 2x ⇔ 1ωn

2!!y + 2 ζ

ωn

!y + y = Kx

Natural Frequency

ωn2 =

1400

⇒ ωn =1

20radsec

%

&'(

)*= 0.05 rad

sec

%

&'(

)*

Damping Ratio

2ζωn

= 20 ⇒ ζ =20 ⋅ωn

2=

20 ⋅ 120

2=

12= 0.5

Static Sensitivity

K = 2

Resonance Frequency

ωr = 1 − 2ζ 2 ⋅ωn = 1 − 2 12

-

./0

122

⋅1

20=

12

120

= 0.035 radsec

%

&'(

)*

Name: ____________________________ HW ID: _____________

Page  3  of  15  

Problem 1 (Continue) (B) (14 Points) For the shock absorber in (A), compute the steady state response for an input displacement

of x(t)= 2+0.2cos(0.1t +10°) . (You should not need to solve the differential equation)

G(jω) = K

1 − ωωn

#

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&

'(

2)

*++

,

-..+ j2ζ ω

ωn

with ωn =1

20 , K = 2 , and ζ = 1

2

at ω = 0.1 = 110

⇒ωωn

=110

120

= 2

G(j0.1) = 2

1 − 2( )2)*

,-+ j2 1

2⋅ 2

=2

−3+ 2j

⇒

G(j0.1) = 2

(−3)2 + 22=

2

9+4=

2

13= 0.555

!G(j0.1) = −tan−1 2−3

#

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&

'( = tan−1 2

3

#

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&

'( = −2.554 rad[ ] = −146.31°

Steady state response:

y (t ) = 2 ⋅ G(j0) +0.2 ⋅ G(j0.1) ⋅ cos 0.1t + 10° +!G(j0.1)( )

= 2 ⋅ 2+0.2 ⋅ 2

13⋅ cos(0.1t + 10° − 146.31°)

= 4+0.111 ⋅ cos(0.1t + 10° − 146.31°) = 4+0.111 ⋅ cos(0.1t − 136.31°)

Name: ____________________________ HW ID: _____________

Page  4  of  15  

Problem 1 (Continue) (C) (12 Points) Match the differential equations (E1 to E4) with the Bode diagrams (F1 to F4) below.

E1: 40000!!y+200 !y+100y =1000x E2: 40000!!y+ 4000 !y+100y =1000x E3: 0.1!y+ y =100x E4: !y+ y =100x

F1 F2

F3 F4

E 1 → F 3E2 → F 4E3 → F 2E4 → F 1

or

F 1 → E4F 2 → E3F 3 → E1F 4 → E2

Name: ____________________________ HW ID: _____________

Page  5  of  15  

Problem 2 (40 Points) You are provided with the following step response for a linear system:

(A) (6 Points) What is the damped period Td and the corresponding damped natural frequency ωd ?

From figure:

Td = 2 [sec]

⇒ ωd =2πT

=2π2

= πradsec

$

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()= 3.1415 rad

sec

$

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()

(B) (4 Points) If a step input of 2.5 [V] caused the above step response, what is the system’s static

sensitivity K ?

K =

10 −02.5

= 4 cmV

"

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&'

Name: ____________________________ HW ID: _____________

Page  6  of  15  

Problem 2 (Continue)

(C) (10 Points) What is the overshoot OS and the damping ratio ζ ? (Clearly state what method you are using to estimate the damping ratio)

OS = yMAX − yfinal = 18 − 10 = 8 [cm]

Damping Ratio:

Use Overshoot method:

ζ =1π

lnyfinal − yinitial

OS

$

%&

'

() =

1π

ln 10 −08

$

%&

'

() = 0.071

Use log decrement method:

ζ =1

2π⋅

1n⋅ ln

Δi

Δi+n

$

%&

'

() =

12π

⋅13⋅ ln 18 − 10

12.1 − 10

$

%&

'

() =

1.3386π

= 0.071

Name: ____________________________ HW ID: _____________

Page  7  of  15  

Problem 2 (Continue) You are provided with the following Bode plot of a system:

(D) (4 Points) Estimate the order of the system. Clearly state your reasoning. 2nd order system

- Phase dropped by -180° - High frequency roll-off at -40 dB/decade

Name: ____________________________ HW ID: _____________

Page  8  of  15  

(E) (8 Points) From the above Bode diagram, assume a second order system then estimate the static sensitivity or static gain K, the natural frequency ωn, and the magnitude at the resonance frequency ωr .

From Bode diagram:

K = 20 dB = 10

At -90° phase angle, ωn = 10 radsec

"

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&'

At pean magnitude, G(jωr ) = 40 dB = 100

(F) (8 Points) Estimate the damping ratio ζ. (Clearly state what methods are you using to estimate the

damping ratio)

Use amplification method:

12ζ

≈G(jωn )

G(j0)=

10010

= 10 ⇒ ζ ≈1

20= 0.05

Other methods are possible

Name: ____________________________ HW ID: _____________

Page  9  of  15  

Problem 3 (40 Points) (A) (10 Points)

ωω

jjG

01.0110)(1 +

= ω

ωj

jG005.011)(2 +

=

x(t) Filter v1(t) Amplifier v2(t) Zin = 105 Ω Zin = 1000jω Ω Zout = 500 Ω Zout = 100 Ω

What is the frequency response of the system relating v2(t) to x(t)? You may leave your answer in the form of a product, rather than expanding and simplifying the total expression.

v2(t )x(t )

=G1(jω) ⋅G1(jω) ⋅L1,2

=G1(jω) ⋅G1(jω) ⋅ZIN ,2

ZIN ,2 +ZOUT ,1

=G1(jω) ⋅G1(jω) ⋅ 1000jω1000jω +500

=10

1+0.01jω⋅

11+0.005jω

⋅1000jω

1000jω +500

Name: ____________________________ HW ID: _____________

Page  10  of  15  

Problem 3 (Continue) You are asked to analyze the loading between the two circuits shown below:

(B) (7 Points) Find the output impedance of G1(jω)

ZOUT is the parallel connection of the inductor and resistor:

ZOUT ,1 =1

1Ljω

+1

R1

=R1Ljω

Ljω +R1

(C) (7 Points) Find the input impedance of G2(jω)

ZIN ,2 is the parallel connection of the capacitor and

the series connection of the two resistors:

ZIN ,2 =1

11Cjω

+1

R2 +R3

=1

Cjω +1

R2 +R3

=R2 +R3

(R2 +R3)Cjω + 1

Name: ____________________________ HW ID: _____________

Page  11  of  15  

Problem 3 (Continue) (D) (6 Points) Write down the loading term between the two stages. If only DC voltage signal will be used

as Vin(t), will you need to worry about loading between the two circuits?

Loading term

L1,2 =ZIN ,2

ZIN ,2 +ZOUT ,1

=

R2 +R3

(R2 +R3)Cjω + 1R2 +R3

(R2 +R3)Cjω + 1+

R1LjωLjω +R1

At DC, ω = 0 ⇒ ZOUT ,1 = 0 and ZIN ,2 = R2 +R3

L1,2 =R2 +R3

R2 +R3

= 1 ⇒ No need to consider loading at DC

Name: ____________________________ HW ID: _____________

Page  12  of  15  

Problem 3 (Continue)

(E) (6 Points) Derive the frequency response function for the above op-amp circuit.

Vout

Vin

= −ZFB

ZIN

= −R

Ljω

(F) (4 Points) What is the mathematical operation this circuit is to perform, i.e. the mathematical

relationship between the input Vin and output Vout ? (Hint: write down the corresponding differential equation between input and output)

Vout

Vin

= −ZFB

ZIN

= −R

Ljω

⇒ Ljω ⋅Vout = −R ⋅Vin

⇒ L ddt

Vout = −R ⋅Vin

⇒Vout = −RL

Vin ⋅dt∫

This is an integrator

Name: ____________________________ HW ID: _____________

Page  13  of  15  

Problem 4 (30 Points) The following signal has been measured (the horizontal axis is time in second)

(A) (6 Points) What is the period T and fundamental frequency in Hz and rad/sec?

T = 8 [sec]

⇒ ω1 =2πT

=π4

radsec

$

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()= 0.7854 rad

sec

$

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()=

18

Hz[ ] = 0.125 Hz[ ]

(B) (4 Points) One of the following expressions is the correct Fourier series representation of this signal.

Choose the correct one:

I. 3+ Bk sin k 2πTt

"#$

%&'

k=1

∞

∑

II. 4.5+ Ak sin k 2πTt

"#$

%&'

k=1

∞

∑

III. 3+ Ak cos k2πTt

"#$

%&'

k=1

∞

∑

IV. 4.5+ Ak cos k2πTt

"#$

%&'

k=1

∞

∑

Name: ____________________________ HW ID: _____________

Page  14  of  15  

Problem 4 (Continue)

The phase shifted cosine series spectrum (the Mk, θk spectra) of a signal is shown below (the phase spectra is in radians):

(C) (2 Points) Write down the fundamental frequency of this signal in rad/sec.

ω1 = 10 rad

sec

"

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&'

(D) (8 Points) Write down the Fourier series representation of this signal.

x(t ) = 2+ cos(30t −0.44)+ 3cos(40t −0.22)+ cos(50t −0.44)

Name: ____________________________ HW ID: _____________

Page  15  of  15  

Problem 4 (Continue)

The (Mk, θk) spectrum of a periodic signal is shown below in (a) and (b). This signal is passed through a filter whose frequency response function is shown below in (c) and (d).

(a) Mk vs Frequency, (b) θk vs Frequency, (c) Gain of filter, (d) Phase of filter

(E) (10 Points) Based on the figures (a) through (d), write down the Fourier series representation of the steady state response coming out of the system. (Note the frequency unit is in Hz and the gain and phase plot are in linear scales)

Input:x(t ) = 1+ cos(40πt − 135°)+0.5cos(120πt − 90°)+0.3cos(200πt −45°)

Output:

x(t ) = 1 ⋅ 1+ cos(40πt − 135° + 90°)+0.5 ⋅0.7 ⋅ cos(120πt − 90° − 180°)+0.3 ⋅0 ⋅ cos(200πt −45°)

= 1+ cos(40πt −45°)+0.35cos(120πt − 270°)

(a) (b)

(c) (d)