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Unformatted text preview: ME563 – Fall 2011
Purdue University
West Lafayette, IN Homework Set No. 10
Assignment date: Friday, November 4
Due date: Friday, November 11
Please attach this cover sheet to your completed homework assignment. Name
PUID Problem 10.1
Problem 10.2
Problem 10.3 TOTAL Homework 10.1
A ﬁxedﬁxed string having a tension of P and mass/length ρ is excited by a force/length of f (x, t) =
f0 g (t), as shown below with f0 being constant in both x and t. If g (t) = sinΩt, the particular
solution for the EOM for this system at x = L/3 can be written as
u(L/3, t) = U (Ω)sinΩt
a) Determine an expression for U (Ω) using modal uncoupling. This result will be in terms of an
inﬁnite summation.
b) Identify the modes that do NOT contribute to U (Ω).
c) Make a hand sketch of U (Ω) vs. Ω for Ω up through the ﬁrst four resonance frequencies that
appear in the response at x = L/3. (Note that not all modes will contribute to this response.)
Clearly indicate all frequencies of resonance and antiresonance in this frequency range. () f0 g t () u x ,t x=0 x = L /2 x =L Homework 10.2
Consider the singleDOF model for a suspension system for an automobile moving with a constant
speed v along a roadway with a spatiallyvarying harmonic function of vertical roughness. It is
your task to design a suspension system with the following criteria:
• The amplitude of the vertical motion of the body no greater than 0.05y0 for a design speed
v = vdesign .
• The transmissibility of the vertical force to the body is no greater than 1.1 for an automobile
speed that corresponds to the undamped natural frequency of your suspension design.
Ignore gravity in your dynamic design. Assume that the following parameters are known: m, y0 , λ
and v = vdesign . Your design will constitute appropriate values for c and k in terms of these known
parameters.
HINT: Recall from lecture that the transmissibility for displacement and transmitted force in a
baseexcited system (such as this system) are the same. You need not rederive the transmissibility
function; you may use the results directly from lecture once you have derived the system’s EOM. v m () k yz c x z y0 ! Homework 10.3
An undamped singleDOF has a base excitation of z (t) = z0 sinΩt. Unfortunately, an operating
excitation frequency of Ω = Ωoper produces resonance. You are asked to design an absorber system
of the type shown below right under the constraint that M ≤ 0.2m.
a) Determine an appropriate set of values for M and K for which perfect vibration absorption
occurs Ω = Ωoper and with the constraint that you achieve a maximum possible separation
between the resulting natural frequencies of the system with the absorber. Express your design
results in terms of m and k of the original system.
b) What is separation between the resulting natural frequencies of the system with your absorber
design? () () x zt k y x zt k m original system K m K
M system with absorber ...
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