ME 36500 EXAM #1 Tuesday, October 7, 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: ___Solution________________

Instructions

(1) This is a closed book examination, but you are allowed one 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  12  

Problem 1 (40 Points) You have developed a novel optical detector that converts incoming light intensity into voltages. The calibration test data shows the following relationship. The cross marks are from the calibration experiments and the straight line is the best linear approximation.

(A) (5 Points) Write down the input range and span (Remember the proper units)

Input Range: [0, 1.8] (mW/cm2) Input Span = 1.8 – 0 = 1.8 (mW/cm2)

(B) (5 Points) Write down the output range and span (Remember the proper units)

Output Range: [0.5, 5] (V) Output Span = 5 – 0.5 = 4.5 (V)

(V)

Name: ____________________________ HW ID: _____________

Page  3  of  12  

Problem 1 (Continue) (C) (10 Points) The best linear fit of the data is shown. What are the sensitivity and the bias? (Remember

the proper units)

(D) (10 Points) What is the maximum nonlinearity as a % of full-scale deflection? What is the input

intensity where this occurs?

Sensitivit:

K =ΔOΔI

=0.50.3

=53= 1.67 V ⋅cm2

mW

#

$%

&

'( =

53

V ⋅cm2

mW

#

$%

&

'(

Bias:b = 0.5 [V ]

Maximum nonlinearity: 5 − 3.5 = 1.5 (V )As percent of f.s.d:

Max(NL)Ouput Span

× 100% =1.5

5 −0.5× 100% =

1.54.5

× 100% = 33.33%

Occurs at input = 1.8 mWcm2

#

$%

&

'(

Name: ____________________________ HW ID: _____________

Page  4  of  12  

Problem 1 (Continue) (E) (10 Points) If the output resolution of the data acquisition system is determined as 0.001 V, what will

be the minimum intensity variation that can be detected by the new sensor?

ΔOK

=0.0015 3

=0.003

5=

35000

= 0.0006 mWcm2

"

#$

%

&'

Name: ____________________________ HW ID: _____________

Page  5  of  12  

Problem 2 (40 Points)

(A) (10 Points) An 8 bit ADC with a nominal range of ±5 volts is given a constant input voltage –2.75 volts. If positive code (unsigned integer) is used, what code will be generated?

(B) (10 Points) If the ADC in part (A) generates a code of 120, what is the range of possible input

voltage that could have created this code? (Remember the units)

Q =2R2N

=5 − (−5)

28=

10256

= 0.0390625 ≈ 0.0391

Code = Round VIN −VADC min

Q

#

$%

&

'( = Round −2.75 − (−5)

0.039

#

$%&

'(= Round [57.6] = 58

code 0 =Voffset = −5 [V ]

Q = 0.0391 V[ ] ⇒ Quantization error = Q2= 0.01955 [V ]

V! IN = code ⋅Q +Voffset = 120 ⋅0.0391+ (−5) = −0.308 [V ]

Range of possible input = V! IN ±Q2= −0.308±0.01955 [V ] = [−0.32755, −0.28845] [V ]

Name: ____________________________ HW ID: _____________

Page  6  of  12  

Problem 2 (Continue) (C) (10 Points) A measurement system has no sample and hold connected in front of the ADC. Calculate

the maximum allowable aperture time for a 10 bit ADC if the maximum incoming signal frequency is 500 Hz and all bits are significant, assuming the signal uses the full range.

(D) (10 Points) At what frequency will a 250 Hz sine wave appear if the sample rate is 140 samples per

second?

f = 500 [Hz]N = 10

ta <1

2N ⋅ π ⋅f=

1210 ⋅ π ⋅500

=1

1.608× 106= 6.217× 10−7 [sec]

= 0.6217× 10−6 [sec]= 0.6217 [µ sec]

fS = 140factual = 250

fapparent = factual − k ⋅fS = 250 − k ⋅ 140 = 30 [Hz]

when k = 2

Name: ____________________________ HW ID: _____________

Page  7  of  12  

Problem 3 (40 Points) (A) (10 Points) Suppose the population of a set of measurements is known to have the following

relationship:

f (x)=Cx if 5 ≤ x ≤ 70 if x < 50 if x > 7

"

#$

%$

Find the value for the constant C that allows function f(x) to be used as a probability density function.

(B) (10 Points) Your boss asks you to establish tolerances for the idle speed of a particular diesel engine model. As a starting point, she suggests you set limits that can be met by 97% of the units coming off the assembly line. Production data indicates a mean value of 500 rpm, and a standard deviation of 20 rpm. Assuming a normal distribution for the idle speed, what are the upper and lower limits you place on the engine’s acceptable idle speed?

As a pdf,

f (x)−∞

∞∫ ⋅dx = 1 ⇒ Cx

−∞

∞∫ ⋅dx = Cx

5

7∫ ⋅dx = 1

2Cx2

5

7

= 1

⇒12

C ⋅ (72 −52) = 1 ⇒12

C ⋅ 24 = 1

⇒ C =112= 0.083

ω = 500 [rpm]Sω = 20 [rpm]

For 97%, use z-table: z97 = 2.17⇒ ω ±z97 ⋅Sω will contain 97%⇒ 500± 2.17 ⋅ 20 = 500±43.4 [rpm] = [456.6, 543.3] [rpm]

Name: ____________________________ HW ID: _____________

Page  8  of  12  

Problem 3 (Continue) (C) (10 Points) You collected 81 readings for the pressure drop (ΔP) occurring across a shut-off valve,

and computed a sample mean of ΔP = 2 MPa, along with a sample standard deviation of SΔP = 0.09 MPa. What is the 99.7% confidence interval (CI) you would place on the true pressure drop? (Remember the units)

ΔP = 2 [MPa]SΔP = 0.09 [MPa]N = 81

99.7% confidence internval = ΔP ±z99.7 ⋅SΔP

N

SΔP

N=

0.09

81= 0.01

N = 81 > 60 use z-table: z99.7 = 2.97

⇒ 99.7% confidence internval = 2± 2.97 ⋅0.01 = 2±0.0297 [MPa]= [1.9703, 2.0297] [MPa]

Name: ____________________________ HW ID: _____________

Page  9  of  12  

Problem 3 (Continue) (D) (10 Points) Since you cannot measure the kinetic energy (KE) of a roller coaster car directly, you

settle for measuring its mass (m) and velocity (v). You determined that m = 500 kg and v = 20 m/s. Knowing that

KE = 12mv2 ,

what uncertainty will be present in your computations of KE if the uncertainty in your mass measurement is 0.3 kg and the uncertainty in your velocity measurement is 0.008 m/s? (Remember to include the unit in your answer)

m = 500 [Kg] v = 20 [m /s]Δm = 0.3 [Kg] Δv = 0.008 [m /s]

ΔKE = ∂KE∂m

⋅ Δm$

%&

'

()2

+ ∂KE∂v

⋅ Δv$

%&

'

()2

∂KE∂m

= 12

v 2

m,v=

12⋅ (20)2 = 200

∂KE∂v

=mv mv = (500) ⋅ 20 = 10,000 (= 104)

⇒ ΔKE = ∂KE∂m

⋅ Δm$

%&

'

()2

+ ∂KE∂v

⋅ Δv$

%&

'

()2

= 200 ⋅0.3( )2 + 104 ⋅0.008( )2

= (60)2 + (80)2 = 10,000 = 100Kg ⋅m2

s2

+

,-

.

/0

= 100 [N ⋅m]

Name: ____________________________ HW ID: _____________

Page  10  of  12  

Problem 4 (30 Points) (A) (10 Points) Match each differential equation (A to D) with the corresponding step response (R1 to

R4) on the right.

A: 0.2 !y+ y =10x B: 0.1!y+ y = x C: 0.01!!y+0.1!y+ y = x D: 0.5 !y+ y = x

A ⇔ R3

B ⇔ R1

C ⇔ R4

D ⇔ R2

0 0.5 1 1.5 2 2.5 30

0.5

1

Step Response

Time (sec)

Am

plitu

de

0 0.5 1 1.5 2 2.5 30

0.5

1

Step Response

Time (sec)

Am

plitu

de

0 0.5 1 1.5 2 2.5 30

5

10

Step Response

Time (sec)

Am

plitu

de

0 0.5 1 1.5 2 2.5 30

0.5

1

Step Response

Time (sec)

Am

plitu

de

R1

R2

R3

R4

Name: ____________________________ HW ID: _____________

Page  11  of  12  

Problem 4 (continue) (B) (10 Points) An accelerometer be modeled by the following differential equation between the input

acceleration (a) in m/s2 and the output voltage (V) in volts: !!V + 70 !V +2500V =10000a .

Find the natural frequency (ωn), damping ratio (ζ), static sensitivity (K) and comment on whether the accelerometer is under-, over-, critically damped. Remember to include the proper unit.

convert to standard form: 1ωn

2!!V +

2ζωn

!V +V = K ⋅a

!!V + 70 ⋅ !V + 2500V = 10000a

⇒1

2500!!V +

702500

!V +V =100002500

⋅a

⇒1

(50)2!!V +

7(50) ⋅5

!V +V = 4 ⋅a = K ⋅a =1ωn

2!!V +

2ζωn

!V +V

⇒ 2 ⋅ζ = 75

, ωn = 50 , K = 4

⇒

ωn = 50 [rad s]

ζ =710

= 0.7 → underdamped

K = 4 [V ⋅s2 / m]

Name: ____________________________ HW ID: _____________

Page  12  of  12  

Problem 4 (continue) (C) (10 Points) A thermocouple is characterized by the following differential equation:

!y+20y = 60x Compute the steady state response of the thermocouple for a sinusoidal temperature input of x(t)= 5+ 3sin(30 ⋅ t) .

!y + 20y = 60x ⇒1

20!y + y = 3 ⋅x

⇒ Frequency response function: G(jω) = 31

20jω + 1

input x = 5+ 3sin(30 ⋅t ) ⇒ ω = 0 and 30

G(j0) = 3, "G(j0) = 0

G(j30) = 3

3020

$

%&

'

()2

+ 1

=3

134

=6

13(= 1.664)

"G(j30) = −tan−1 (τω) = −tan−1 3020

$

%&

'

() = −tan−1 3

2

$

%&

'

() = −0.9828 [rad ] = −56.3°

⇒ yss (t ) = 5 ⋅3+ 3 ⋅ 6

13⋅ sin(30t −0.9828)

= 15+4.9923 ⋅ sin(30t −56.3°)

= 15+ 3 ⋅ 6 1313

⋅ sin(30t − tan−1 32

$

%&

'

())