Exam1_F09_sol

# Exam1_F09_sol - <1 0 to ’T 1—0 Name 2 Section...

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Unformatted text preview: <1 0 to ’T 1—0- Name: 2 Section: 9:30_Savran 11:30_Ariyur 1:30_King ME 365 EXAM 1 Thursday October 8, 2009 6:30-7:30 p.m. Lilly 1105 POINT DISTRIBUTION SCORE PROBLEM 2 3 5 points 35 points 40 points 40 points 150 points Percentage: 2/3 Total = ' Don't forget your name and please circle your section instructor. ° Exam is 7 pages including this cover page: make sure you’re not missing any pages. 0 [f you use extra pages, indicate on the problem page that you’re continuing onto an extra page. ' Pay attention to units. ' Explain your reasoning. Correct answer with wrong explanation gets no credit. Problem 1. A MEMS (Micro Electro Mechanical System) accelerometer used in inertial measurement systems on artillery shells is supplied with the foilowing specifications: Input range: —320009 to 320009 (where 9 is the acceleration due to gravity, and 9:9.81m152) Bias=0.1 mm Sensitivity: 0.01 pmlg Output ran9e=—0.22mm to 0.42mm Maximum noniinearity=1% FSD (Full Scale Deflection) at +160009. (The actual nonlinear response exceeds the linear prediction.) (Input to the system is in 9’s and the output is mm’s) a. What is the resoiution of the accelerometer if the smallest displacement we can measure is 1 nm (10'9 m)? AI; AO/K : iquM {3A 3 W Mtg : b. What is the expected displacement output of the sensor at an acceleration of 100009? Assume that nonlinearity is negligible at this acceleration. it ~ Li 0* Kitty: {OWQMIEEXiOQSiJG‘lM : 2M0 m:<9.2_mm c. What is the value of the maximum nonlinearity in output units (e.g., pm)? No F-So a; c.0ixtoiqz_—Ga;®mm a whim Mum d. What is the actual displacement output when the acceleration is 160009? Why? 8‘ ,e mi; _ tto \téiigglf; m ~: WW“ Problem 2. You have an 8 bit analogue to digital converter (ADC) with an nominal input range of i 5 volts. (a) Determine the quantization interval (Q) of the ADC in volts. 8:: l0. Oquv My 2} (b) What is the maximum quantization error? : O‘DZV .y- v#_ 2, (0) Determine the output code in decimal and binary to an input of —3.7 volts assuming all codes are positive integers. 33': 2.? , , l 4/ Carla: (Basel «7933;; : .71.; r; (r5;- :31 0000 s) (9.1: 3"? 3' (d) Assuming the ADC has an aperture time of 350 nsec. (3.5 e — 7)sec., determine the effective number of bits of resolution to an input signal of 10 kHz. 2“ \ w ' “I dial (7:) n 43?, :61 679-535“ : .w «— ~----'-" -0 _ . Tl!” *‘ 7’? T "l - - ta. 1 35m new a, 6...; my (e) lf this 10 kHz signal is sampled at 7,500 (Samples I sec. ) determine the apparent frequency of the sampled signal. 5": . , ..-»\ - \ Em ; 1.3“ Lilac 1. 9/ . __\>>_— Mu..-ﬁ. .. 7 u- ,- (f) What effect if any will this apparent frequency (part e) have on the effective number of bits of resolution (part d)? State your reasoning. . F, . ’\ 4 . i ”( [4:3 ﬂfrgf a)“ r“ «g. ,n. x. i 1‘ ,v " GEM“ «. mmmﬁﬁ 93ml 6 6W3; Problem 3. Part A. Below is the probability density function for selling a car based on how old it is. W) a 40 x (year a) What is the value of a? 1X79; 5:3“ x + Q 2.19 (if) _ ‘ . x u. )— ~ a ' rm r'\ x , W r: a“; wti :l :ars07~®+ AW; \$.13, :40 2— Lie 2, 200‘ I CAj‘l/ao 1': @rDtE/wfjﬁ b) What is the mean age for selling a car? go “u l r ,p. 3 Im 2— ' M, «« SKQG) dx : (“3,3353 fl +Q‘ebx (\$71 : “wig + €2,603; \ f7— ” Lilo qa 3 3- is .3 O c) if you have just bought a brand-new car, what are the chances that you will sell it within the first 2 years? . .7 i 2<- ' ﬂ '“dr :2— ..n ,...,_ l 17(% L2) "'3 J yi+3c3ﬁ_ -_: -' if” 3‘“ 0.,«5‘237- f : _{j}£’3 in gro'ggfl C) Lt C3 2. K ‘3 O 2' : 033‘fo 0r” w “it”; d) What is the probability of selling a car that is older than 45 years? “if; it"? Part B. You measure the length of 100 rods that will be installed in cars. You ﬁnd a mean of 50 mm with a standard deviation of 5 mm. a) What is the 90 % confidence interval? 5 0mm “t 3:6; 5 Z ‘50: “"33 1“ :ﬂgg mi" ’” ‘3';— {a i id??? b) How many more measurements do you need for the same interval in part a to be a 95 % conﬁdence interval? (assume that the mean and the std. dev. don’t change). ’ I? 3‘ Ci; 5' {ASA :1 *L‘ijiiw‘ 2 1‘ ; Car—ﬁr lp “N Problem 4. A system’s behavior is modeled by the following ODE: 0.2 3'2 + 2y = 4x a) What are the time constant and the static sensitivity of this system? (ﬂuirygzxgj Ody-Jr‘lril-x' (Let-m; 14:; 2— . b) What islare the eigenvalue(s) of this system? A? I V»- v "{9}! (L H” i) c) What are the natural frequency and the damping ratio of this system? _ ,w: t ~.L _ boa-t east, the e. lj Seder" system d) What is the particular solution due to a unit step input? (“unit step” means a step change of "1"). e) What is the homogeneous solution? a 19”“; : Aer ., ,t{ \1 it I A a f) What is the total solution for zero initial conditions? , visit a ._ ‘- 2 LyA'e CR: “ii-As” 1’“ :rl g) If the time constant of the system was 5—fold larger, how would this affect the particular soiution for a unit step input? 1 ' \ ~. . 930 ‘ar‘x S C4153 5-qce EQMV‘CQLU/ jqiﬂ‘ ’5 CQQJ'LQICC ...
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