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Unformatted text preview: ME 563  Fall 2004
Final Exam
Problem No. 1  20 points The three—DOF system shown below:
2fo sith 5
fo sith has natural frequencies of: col 0.7 rad/sec
wz = 1.0 rad/sec
(03 = 1.6 rad/sec and a mass normalized modal matrix of: 0.3384 0.7242 0.8801  ;_ ~(I) ~(2) ~(3)
[P] 0.5276 0.5832 '0'4673JJE [9 «_p g ] 0.7791 0.3681 0.0835 Find the particular solution for x, corresponding to an excitation frequency of
£2 = 1.3 rad/sec if f0 = 0.40 newtons. E ME 563  Fall 2004
Final Exam
Problem No. 3  20 points Consider a forcing f (t) a 2 sin6t. a) What is the least fundamental period of f( t)? (The least fundamental
frequency is the shortest time over which the function repeats.) b) Determine the Fourier series for ﬂ t). f(t) W t Jt/6 Jc/3 n/Z 2n/3 ME 563  Fall 2004
Final Exam
Problem No. 4  20 points An undamped singleDOF system:
mjr' + kx  f (I) is given the excitation f( t) shown below: f(t) e
2fo
fo
T 2T t
W for the re5ponse of the system corresponding to initial conditions of x(0)  x0 and x(0)  0. Clearly indicate the
solution that is valid for the time ranges of: 0 OstsT
° TstsZT
° r22T You are NOT asked to evaluate these integrals. ME 563  Fall 2000 Name Final Exam
Problem No. 3 A damped threeDOF system has the following set of EOM’s; [MB + [CB +[K]x  fusith (0! 3L0
—1 l J [C]  BIK] where and [i = 0.1 sec. The undamped natural frequencies (0 (j = 1,2,3) and massnormalized modal vectors l  1,2,3) have been determined, with the following results: 001  1.85rad/ sec 002  4.24rad/ sec
(03 : 4.85rad/ sec 0.372 0.707 0602
if”  [0.6021/ $73 in)  l 0 l/ JR; é”) =l0.3721/JE
[0.372] [—0.707] [ 0.602 J For this problem: a) Write down the uncoupled modal differential equations of motion. In these
equations, explicitly indicate the modal damping ratios, the modal natural
frequencies and the modal forcings. b) If the particular solution of the original EOM’s is written as:
xpilt)  Disin(S2t  ¢i) ; j = 1, 2, 3 make sketches of D1 and D2 as functions of the exciting frequency 9. c) Interpret your sketch in b) above in terms of the modal contributions to the
response. ME 563  Fall 2000 Name FﬁnalExanI
Problem No. 4 The following undamped, singleDOF system:
x(t) l—’ ft
()
k is given a forcing f(t) shown below and initial conditions of x(O)  x(O)  0. For this system, let k =101r2newtons/ meter and m  10kg . f(t) t (secs) For this system: a) Write down the convolution integral solutions for:
i) t = 1 sec
ii) t = 4 sec
iii) t = 6 sec
DO NOT EVALUATE THESE INTEGRALS. b) Based on the graphical convolution approach, determine the times at which
i) the first five maxima occur in x(t)
ii) the first five minima occur in x(t) Multiple copies of f(t) are provided on the next sheet that you can use to assist you
in making sketches for determining these times for maximum/ minimum
responses (this sketching sheet will NOT be graded). c) Make a sketch of x(t) on the axes provided above for a time range that includes all
of the times listed above in b). ME 563  Fall 2000 Name Final Exam
Problem No. 5 As shown in Figure (i) below. a loading f(x,t) acts on a wing of an aircraft, with the loading
acting over a short period of time T = 0.2 seconds. f(x.t) F(t) F(t)
fo
x = 0 x = L 0 T/2 T g t
Figure (i) Figure (ii) Figure (iii) Suppose that the wing is modeled as a singleDOF system (as shown in Figure (ii) above)
where k = 10,000 newtons / meter, m = 100 kg and the forcing F(t) is as shown above in
Figure (iii) with f0  120 newtons. a) Treating F(t) as a SHOCK loading, determine an upper bound AND a lower bound
on the value of zmax for the response of the singleDOE model shown above. b) Treating F(t) as an IMPACT loading, determine an 112m on the value of
zmax for the response of the singleDOF model shown above. c) Does the wing serve as an effective shock isolator in transmitting the applied force to the support of the wing? (You may need to refer to the lecture notes to recall the
definition of effective shock isolation.) IGNORE THE INFLUENCE OF GRAVITY IN YOUR CALCULATIONS. ME563  Fall 1998 Name
Final Exam
Problem No. 1 The mass matrix [M] and the ﬂexibility matrix [A] ( = [K].1 , where [K] is the
stiffness matrix) is given below for a 5DOF undamped system. a) Determine a lower bound on the lowest natural frequency for the
system. b) Derive the form of the Rayleigh quotient for an undamped system in
terms of its ﬂexibility matrix and mass matrix. c) Using [M] and [A] in the Rayleigh quotient found in b) to find an M
bound on the lowest natural frequency for the system. [10 1 [2 2 2 2 2]
 10 I I2 4 4 4 4 [M]: 10 :(kg) [A] :2 4 6 6 6:x10'4(meter/ N)
 10 I I2 4 6 8 8'
l 10] [2 4 6 8 9] ME563  Fall 1998 Name Final Exam
Problem N o. 3 An undamped, singleDOF oscillator experiences a base motion of y(t), with
y (t) as shown below. The center of mass motion of the homogeneous cylinder is to be described by the coordinate x(t), where x(t) is measured
relative to the base. The oscillator is given initial conditions of: x(0)  5((0)o. a) Write down the convoluti%n integral solution of x(t) for 2a < t < 3a. DO
NOT EVALUATE THIS INTEGRAL. b) Say that we know that a = n V3m/2k. Without evaluating any integrals, make a sketch of the response x(t) for 0 < t < 5a on the axes
provided below. In your sketch, clearly identify the times corresponding to: local maximum of x(t), local minima of x(t) and all zero crossings of x(t).
y(t) x(t) —>‘—>I k m
I !/ no slip y(t) ME563  Fall 1998 Name
Final Exam
Problem No. 5 IQNORE QRAVITY IN YOQR WORK WITH THI§ PROBLEM. a) A constant force f0 = 100 N is applied to a particle of mass M that is supported by a beam of length L and having negligible mass compared to
the particle. As a result of the application of this force, the tip of the
beam deﬂects by an amount of 0.01 meters. If M = 10 kg, what is the natural frequency (on of this system?
f0 I‘— L A El negligible mass A harmonicallyvarying force (with a magnitude f0 = 100 N) is now applied to the tip of the beam with the frequency of the force
(unfortunately) tuned to the natural frequency of the beam/ particle
system. To eliminate the resonant response of the beam, a second
particle B (having a mass of 2M) is attached to the tip of the beam with a
spring of stiffness K. i) What must be the stiffness of this spring in order to completely
eliminate the vibrations in the beam? ii) What is the amplitude of the motion of the newlyattached particle
B? f0 coswt negligible mass ...
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