This preview shows pages 1–8. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 (IJ‘)/\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 ...
View
Full
Document
 Fall '09
 HOWELL

Click to edit the document details