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Unformatted text preview: ME563 – Fall 2006
Purdue University
West Lafayette, IN Homework Set No. 5
Assignment date: Wednesday, September 20
Due date: Wednesday, September 27
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: C.M. Krousgrill
585 Purdue Mall
Purdue University
West Lafayette, IN 479072088 Name
Location _________________________________________
Problem 5.1 __________________ Problem 5.2 __________________ TOTAL __________________ ME 563 – Fall 2006
Homework Prob. 5.1
Consider the threeDOF system shown below whose motion is described by the absolute
generalized coordinates x1 , x 2 and x 3 , where x1 = x 2 = x 3 = 0 when the springs are
unstretched.
! x1
! !x
2 k
k ! x3
k
m m 2m k k a) Determine the mass and stiffness matrices, [M] and [K], for the system.
b) Determine the characteristic equation for the system using the solution form of:
x = X ei"t . ! c) Divide the above characteristic equation by k 3 and rewrite the characteristic
equation in terms of: µ = m" 2 / k . Note that this characteristic equation is
independent of m and k (all coefficients in the characteristic equation have
numerical values).
!
d) Use a nume! cal method to determine numerical values for the three roots of the
ri
characteristic equation above. (You may use the Matlab function “roots” as
detailed in the solution of several lecture examples or any other means available.)
From these, write down the three natural frequencies "1, " 2 and " 3 in terms of
m and k.
e) Find the three modal vectors X (i ) (i = 1, 2, 3) for the problem. Mass normalize
!.
!
˜ (i ) i = 1, 2, 3 !
these modal vectors to produce X ( ) f) Verify the orthogonality of the modal vectors with respect to [K]:
!
˜ (i )T
˜ ( j)
X [K ] X = 0 (i " j ) .
g) Make sketches of the ! odal vectors found above.
m ! h) Find the response due initial conditions of: x1 (0) = x 2 (0) = x 3 (0) = 1 and
˙
˙
˙
x1 (0) = x 2 (0) = x 3 (0) = 0 . Comment on the relative size of the contributions of the
three modes to the response. !
! ME 563 – Fall 2006
Homework Prob. 5.2
Identical particles 1 and 2 (each of mass m) are connected by a spring of stiffness k. A
massless bumper is attached to the right side of particle 2 with a spring of stiffness 2k. A
third particle (also of mass m) approaches the bumper with a speed of v, and on contact,
sticks to the bumper at t = 0. After sticking (t ≥ 0), the absolute generalized coordinates
x1 , x 2 and x 3 are to be used to describe the motion of the system, where
x1 = x 2 = x 3 = 0 when the springs are unstretched. Prior to sticking, particles 1 and 2 are
stationary, and all springs are unstretched.
!!
! ! 3 a) Using T = 3 1
"2U
˙˙
" " Mij xi x j and Kij = "x "x , determine the mass and stiffness
2 i =1 j =1
i
j matrices for after sticking.
b) Show that the stiffness matrix is singular (and therefore, the system has a rigid
!
! body mode).
c) Set up the characteristic equation and determine the three natural frequencies.
d) Find the three modal vectors X (i ) (i = 1, 2, 3) for the problem. Mass normalize ˜ (i )
these modal vectors to produce X (i = 1, 2, 3) .
˜ (i )T
˜ (i )
e) Verify that: X [K ] X = "i2 .
!
f) Find the response of the system for t ≥ 0. In your solution, identify the
!
contribution of the rigid body mode to the response.
! massless
bumper v
1 2 k
m m x1 m x2 x3 k
m 3 2k sticking 2k
m m ...
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 Fall '09

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