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Unformatted text preview: ME563 – Fall 2006
Purdue University
West Lafayette, IN Homework Set No. 9
Assignment date: Wednesday, November 1
Due date: Wednesday, November 8
Please attach this cover sheet to your completed homework assignment.
• If you choose to scan in your completed solution, please email this
solution to vagrawal@purdue.edu by midnight of the due date given
above. Please give your homework file the following name: “name_hwk”.
For example, if your last name is Smith and you are submitting Homework
Set No. 3, then you should name your file: “smith_03”. • Otherwise, if on campus, please bring to lecture on the due date. If off
campus, mail to the address below with a postmark no later than midnight
of the due date given above: Ms. Donna Cackley
585 Purdue Mall
Purdue University
West Lafayette, IN 479072088 Name
Location _________________________________________
Problem 9.1 __________________ Problem 9.2 __________________ Problem 9.3 __________________ TOTAL __________________ ME 563 – Fall 2006
Homework Prob. 9.1
Here we consider a twoDOF model of an automobile traveling on a roadway as shown
below. The automobile is traveling with a constant speed (z = vt) on a roadway whose
roughness is modeled by a sinusoidal function y ( z) = y0 cos( 2"z / #) where λ is the
wavelength of the roughness. The absolute coordinate x is used to describe the vertical
displacement of the center of mass G of the automobile body. The rotation of the
automobile body is described by the rotational coordinate θ. The body is to be modeled as
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a homogeneous rigid thin body of length L. Assume that the tires A and B remain in
contact with the roadway; otherwise, ignore gravity in your analysis. λ a) Use the Lagrangian formulation to derive the EOM’s for this small oscillations of
the coordinates x and θ. Use L/λ = 0.5.
# x ( t ) & # X () ) &
b) Solve for the particular solution of the EOM’s: $ P
'=$
' cos )t
% L " P ( t )( % L * () )(
c) Determine the natural frequencies and mode shapes. Use these to write down the
uncoupled modal EOM’s. Solve these modal EOM’s.
d) Using the results from either b) or c) above, make a sketch of X(Ω) and LΘ(Ω)
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vs. " / k / m .
e) From your plots in d), identify speeds v corresponding resonances and antiresonances for the coordinates. ! ME 563 – Fall 2006
Homework Prob. 9.2
An undamped, singleDOF system has an EOM of the form:
m˙˙ + kx = f ( t ) = f 0 sin "t cos3 "t
x ! a) Observe that the “forcing” f ( t ) is a periodic function of time. Using trig
identities, determine the Fourier series for f ( t ) . Identify the fundamental period T
of the forcing (the fundamental period is the shortest time interval over which the
function will repeat, f ( t ) = f ( t + T ) ).
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b) At what frequencies " are resonances expected in the response of the system?
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c) Determine the total response (homogeneous + particular) of the system
˙
corresponding to initial conditions of x (0) = x 0 and x (0) = 0 .
!
! ! ! ME 563 – Fall 2006
Homework Prob. 9.3
An undamped, singleDOF system is given a base motion of ˙˙ ( t ) = a0 f ( t ) where
y
f ( t ) = f ( t + T ) is the Tperiodic function shown below.
x
y ! ! m k f(t)
1 t
1
T T T T T T From lecture, we know that f ( t ) = f ( t + T ) can be written in terms of its sinecosine
Fourier series as:
# # f
f ( t ) = 0 + $ f cj cos+ $ f sj sin j"t
2 j =1 !
j =1
where " = 2# / T . ! !
!
! a) Considering the Fourier series for f(t) above, provide arguments that:
f0 = 0
f cj = 0 ; j = 1, 2, 3,...
and
T /2 f sj = 2 # f (t ) sin j"t dt ; j = 1, 2, 3,... 0 b) Considering the results of a) above, determine the Fourier series for f(t). You will
likely need to use integration by parts to evaluate these integrals.
! ...
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 Fall '09

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