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MidtermSolutions - UNIVERSITY OF MICHIGAN Official...

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Unformatted text preview: UNIVERSITY OF MICHIGAN Official Examination Booklet Name of Student: SOL-“Tl OIL-l S Date of Examination: March 15, 2006 Course: CEE511 — Structural Dynamics The University of Michigan Honor Code The Honor Code outlines standards for ethical conduct for graduate and undergraduate students, faculty members, and administrators of the Coiiege of Engineering at the University of Michigan. Policies When Taking Exam The instructor need not monitor the exam The instructor will announce the time of the exam and the instructor's whereabouts will be communicated to the class Students will allow at least one empty seat between themselves and their neighbor The instructor will inform the class prior to the exam if aids are aEEowed during the exam Students must write and sign the Honor Pledge on their exams The Honor Pledge is: I acknowledge that I have neither given nor received aid on this examination nor have I concealed any violation of the Honor Code. (Signed) QUESTION 1: EQUATIONS OF MOTION OF A MACHINE FLY-WHEEL (15 pts) a. PIease formulate the equation of motion for the system shown below: u(t) In this system, the small mass, m, is attached to a fly-wheel as shown. The fly—wheel with radius, R, essentially pivots about the point “0”. The wheel with inertia, [0, is held in equilibrium through the spring, k, and damper, c. Write the equation of motion in terms of degree—of—freedom, u, as shown. Assume in this problem that the dead weight of the hanging mass has already statically displaced the mass prior to a verticai load, pfl) being applied to the hanging mass. b. What is the natural period of the system? (Write it in terms of the unknown variables shown above) 0. Write an expression for critical damping of the system (Write it in terms of the unknown variables shown above) QUESTION 2: APPLIED LOADING ON STRUCTURE (30 pm) For a simple single degree—of—freedom system defined by a mass (m) and spring (k), please write an analytical expression describing the response of the system, ufl) to the following load, 13(13): The first 0.75 seconds of the loading, pm, is essentially a sinusoidal loading (with a period of I second) on the system, as shown. A's 3% = A gimme.._15:¢§2:r123c.. . " ' ' —4w'2.A“g~mg;wsW 2.1% . 4ffiz'm gram 4 4#2M'3'@'2'nae wm m «Lgi'aejm. + m pa-émmé) "'<z«—‘r%r*tm)Asy'h;+r'b'+ (b.4w2-eqse'4gm'ucce pa.-?,@gm .5. .. a“ sew o1 93 § i #0 f; 1 . .. . j; . : _ .: kififiamfli .. k; g i I . . <:. “:3. E WWW mama _ i _ i _ _ .. . . I ._ . . .I I. .I I. . . .Ek.g*§figbfi:k: : E . :.. . . 160 SM EETS 200 SHEETS 22-1 44 50 SHEETS 22-142 22-144. #5? , dfi$fifififi [email protected]%#e% icfl2x£¢gg% E+fimp¢&Q$;f1- : k-frflm E 3 .K -.....j-&g§.vfl"§f?mé§_.. . i - -- + %'“'PO 5 k %figm .64; 25:35 $- gawfitgfi;;ghmp1%4;gggj;3" *iJ l i E i .LxLCoJS): -- ME. ME 4311“ M - i5?- - “ n m u m n M. m . --:m.im----+ 4-. _ . let go and free to vibrate. Please plot a graph history of the tower if we assume the structure to f—freedom system. 011 that graph, please note any “key” t. a a _ . » . . _ IIIIriI I I I III I _ . . _ _ _ . . _ _ _ . _ . _ _ _ . . . . _ _ _ . LII. . rcLluL TlLl \II-I C e 3 II... :0 a r ( VJ C _ I n --..v---+---+-- --J_.:.I--.Ju J.----T--r--. W _ _ — w _ . w..- IS a . _ a _ m _ . _ S e um 3 O cowmoEcmmE 289:5 The Lurie Tower is instrumented with a variable shaker to excite the tower at different harmonic frequencies. Based upon the steady state response of the Lurie Tower to the harmonic excitations, you are able to determine the following results: Let us now assume the Lurie Tower can be given an initial displacement, say 4 cm. Once QUESTION 3: SYSTEM I.D. AND PREDICTED RESPONSES (20 pts) achieving this displacement showing the resulting Vibration time behave as a single degree- ; also, if the response exhibits a , please calculate the first four peaks (response magnitude). points, such as zero crossing (times at which they occur) harmonic like response §§§§§§§§E§¥2§§§2§E 9%.? r411: cosmoEcmmE 289:5 T5511. .--.L ____L i r ‘ 1 Frequency (rad/sec) 539’“ l I “Hf/#4. 4T "g M: 3’ Vl-égkflog onguéfi . __ mm 3 .3 3 i 5 z QUESTION 4: LABORATORY TESTING FOR SYSTEM PROPERTIES (15 pts) In the laboratory, you test a steel frame structure under a cyclic load patter. You expeu'mentally observed the following force-deflection curve for the frame structure. Find the viscous damping coefficient, 5, and the logarithmic decrement, 6, of the tructure. S {Assume w:wfl> Force Force (kips) Deflection (in) ) Deflection (in) I lllPl I I 3%: $20”. wfiwmm mam EVEN 3m E wimmw 92. $2M ‘Emmamhmwmz 35% am Eumm 6 term. gut); pct) 6 At QUESTION 5: GENERAL QUESTIONS (20 pts) a (6 pm) (i pts) (3 pts) 0 If the impulse response of a system is equal to: : e—t/w“ Please write an analytical expression for the response of a single degree of freedom system exposed to any type of loading, pm. Please provide one numerical integration method that is I I unconditionally stable. A V5 (4625 A66?!— — Mewfikfi-K- ( 3 '7 {2 /5 3 4) (an (orgasm Acwu— Ham-mm: (ZS-.0 / 3 :0) Please list three common types of accelerometers employed to monitor structures. MFAotfiVfi , F9202. 3M. J, Plguem‘t In the design of an accelerometer, what is the typical value for the damping ratio of the sensor? f: o n '7 Draw the shape of a typical pseudo—acceleration design spectra as a function of structural period, T". Please draw two spectra on the same plot with one spectra at 1% and the other at 10% critical damping. ...
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