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Unformatted text preview: ME563 – Fall 2011
Purdue University
West Lafayette, IN Homework Set No. 9
Assignment date: Friday, October 28
Due date: Friday, November 4
Please attach this cover sheet to your completed homework assignment. Name
PUID Problem 9.1
Problem 9.2
Problem 9.3 TOTAL Homework 9.1
Consider the EOM for a forced, damped singleDOF system:
2
x + 2ζωn x + ωn x =
¨
˙ f ( t)
m Starting with the general form of the convolution integral solution derived in lecture (see equations
(IV.12) and (IV.13) in Section IV.3 of the lecture notes), derive the convolution integral for an
underdamped (0 < ζ < 1) system. Note that the ﬁnal form of the underdamped convolution
integral solution is found on page IV36 of the lecture notes; you can use this to check your answer). Homework 9.2
Consider the EOM for a forced, undamped singleDOF system:
2
x + ωn x =
¨ f ( t)
m In this problem, you are asked to use the convolution integral approach to produce a qualitative
sketch for the response of this system with the excitation shown below and zero initial conditions
corresponding to:
a) T = π /ωn
b) T = 2π /ωn
for 0 < t < 4T .
In your sketches, clearly identify the following:
• the exact times for the zero crossing of x(t)
• the exact times and corresponding response values for local maxima x(t)
• the exact times and corresponding response values for local minima x(t) Homework 9.3
Consider the cantilevered beam shown below where the beam has negligible mass and a ﬂexural
rigidity of EI . Three particles, each of mass m, are attached to the beam at locations shown. The
transverse deﬂections for the particles will described by the generalized coordinates y1 (t), y2 (t) and
y3 (t). A transverse force F (t) is applied to the beam as shown in the ﬁgure. Although we generally write our EOMs for multiDOF systems in terms of the mass and stiﬀness
matrices:
¨
[M ] + [K ] = f (t)
y
y (1) we will write our EOMs for this problem in terms of the mass and ﬂexibility matrices:
¨
[D] + = [A]f (t)
yy (2) where [D] = [A][M ] and [A] = [K ]−1 = ﬂexibility matrix. We will do so here to take advantage of
some previouslyderived results for the ﬂexibility matrix for this problem.
a) Develop the mass matrix [M ] and the forcing vector f (t) for the (t) coordinates.
y
b) Write down the [D] = [A][M ] matrix. Note that the [D] matrix is symmetric. Do not derive the
ﬂexibility matrix [A]; you may directly use the results of Example I.6.1 from the course lecture
notes.
c) Note that the eigenvalue problem for equation (2) above using (t) = Y e−iωt is:
y
2
−ω [D ] + [ I ] Y =
0 where [I ] is the 3 × 3 identity matrix. Determine the natural frequencies ωj and and modal
ˆ
ˆ
vectors Y (j ) . Normalize the modal vectors such that Y (j )T [D]Y (j ) = 1. d) Using the modal coordinate transformation:
( t) =
y 3
ˆ
Y ( j ) pj ( t)
j =1 write down the uncoupled EOMs for the modal coordinates p1 (t), p2 (t) and p3 (t). Rank order
the magnitudes of the modal forcing functions.
e) (10 bonus points) Suppose that the forcing on the beam is F (t) = F0 δ (t), where δ (t) is the
Dirac delta function. Determine the response of the system (t) corresponding to zero initial
y
conditions. ...
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 Fall '09

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