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test3 - Fall 2011 AAE 340 — Dynamics and Vibrations Exam...

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Unformatted text preview: Fall 2011 AAE 340 — Dynamics and Vibrations Exam III 50 I uﬁons Please read the problems carefully. Write clearly and use diagrams when necessary. MA 35 MOMENTS OF INERT/A OF COMMON SHAPES thin "94 ‘l~ cm on _ mra' I” ‘133 ‘ T x I; = 25°- ‘Hfm diSK "" .. ma} In ” 75; a" .. mba' Isa - *— ( 30 points) 1. Shown below is a yellow thin, ﬂat plate of uniform density. The plate is circular (radius R = 2 met) with a square section cut out of the middle (length of each side is 2x5 met). Let mass density V = 12 kg / me and the cm is at the geometrical center of the plate If unit vectors 1; are ﬁxed in the plate, determine the inertia matrix [10m] . Q‘) b1 ( 40 points) 2. A uniform thin disk (mass m, radius R) is welded on a “massless” shaft. The shaft is mounted in frictionless bearings (at A and B) and attached to a rotating frame F. The shaft rotates at constant angular speed a) ; the frame rotates at the constant rate A. (a) Deﬁne a set of variables to describe the position and orientation of the disk. Determine the number of DOF. Justify your answer. (b) If i inertial unit vectors and l; are ﬁxed in the disk, determine the direction cosine matrix [2 l. t . 3‘ (c) Derive an expression for angular momentum associated with the center of mass of the disk, i.e., i170". Express TV” in terms of unit vectors ﬁxed in the frame F. i (d) For the disk, derive expressions for dt D “C ’C: marital tn game. A La» ' l’ A ‘. fixed; to disk L' L! "59k Cok\$@ CoLCp, ' - , A . A A ’\ 05b= Toﬁ \$631” = 0(‘1Ca, +@» 75: = A'Fa.+ 60151 =/\(c(bb’;—§gb\$\+wb? = Lab“. +A©ﬂ3b§ —-/\s.(be we“; £fo {32“} L W ME +J‘Aw5pb’é. ~M—J‘A’“ (1/515) z (I-J‘)/\a8{50¢ 5‘, + (I— :11) w/\ 3(3) 1921 “IALU (1,5 bi; ( 30 points) 3.-_ Continue with problem 2: (a) Sketch the FBD that represents this problem. (b) Determine the bearing forces at A and B; express the bearing forces in terms of (i) unit vectors [3 (ii) unit vectors ﬁxed in frame F M “ dt 4 {(31%ng + ‘/L(A5’Ba)bg. + Mt: bl = (I—J)A“s¢cg,r§, + (J—ar)w/\sbb3, -JAmé/pg ...
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test3 - Fall 2011 AAE 340 — Dynamics and Vibrations Exam...

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